# CPU-GPU Heterogeneous Code Acceleration of a Finite Volume Computational
Fluid Dynamics Solver

Weicheng Xue  Hongyu Wang  Christopher J. Roy

###### Abstract

This work deals with the CPU-GPU heterogeneous code acceleration of a finite-
volume CFD solver utilizing multiple CPUs and GPUs at the same time. First, a
high-level description of the CFD solver called SENSEI, the discretization of
SENSEI, and the CPU-GPU heterogeneous computing workflow in SENSEI leveraging
MPI and OpenACC are given. Then, a performance model for CPU-GPU heterogeneous
computing requiring ghost cell exchange is proposed to help estimate the
performance of the heterogeneous implementation. The scaling performance of
the CPU-GPU heterogeneous computing and its comparison with the pure multi-
CPU/GPU performance for a supersonic inlet test case is presented to display
the advantages of leveraging the computational power of both the CPU and the
GPU. Using CPUs and GPUs as workers together, the performance can be improved
further compared to using pure CPUs or GPUs, and the advantages can be fairly
estimated by the performance model proposed in this work. Finally, conclusions
are drawn to provide 1) suggestions for application users who have an interest
to leverage the computational power of the CPU and GPU to accelerate their own
scientific computing simulations and 2) feedback for hardware architects who
have an interest to design a better CPU-GPU heterogeneous system for
heterogeneous computing.

###### keywords:

heterogeneous computing , performance model , CFD , GPU

[label1]organization=Department of Computer Science and Technology, Tsinghua
University, city=Beijing, postcode=100084, country=China

[label2]organization=Internet Based Engineering, city=Beijing,
postcode=100083, country=China

[label3]organization=Department of Aerospace and Ocean Engineering, Virginia
Tech, city=Blacksburg, postcode=24060, country=USA

This work proposed a performance model for CPU-GPU heterogeneous computing to
estimate the performance of a finite volume computational fluid dynamics
solver. Also, the performance model can provide some guidance for future
computer architecture design and domain application performance optimizations.

This work applied CPU-GPU heterogeneous computing to accelerate a
computational fluid dynamics test case and compared its performance with a
pure GPU implementation. With a proper workload ratio assigned to the GPU and
CPU workers, the CPU-GPU heterogeneous version outperforms the pure GPU
version by up to 23%, which can be estimated fairly by the heterogeneous
computing performance model proposed in this work.

##  1 Introduction

The increasing demand for high-performance computing from computationally
intensive applications, including scientific computations and artificial
intelligence applications, has greatly driven the development and adoption of
heterogeneous computing. As of June 2023 (the latest release), most
supercomputers in the Top500 list [1] and the Green500 list [2] adopt
heterogeneous architectures, i.e., different types of processors,
accelerators, or specialized hardware. Heterogeneous computing usually
combines the advantages of various heterogeneous components, allowing
application users to flexibly select appropriate compute units to execute
specific computational tasks to satisfy different needs including improving
performance, promoting scalability, saving energy, etc. Mittal et al. [3]
surveyed a lot of heterogeneous computing techniques mainly for CPU-GPU
heterogeneous computing including the architecture, scheduling, data
transfers, programming models, and application domains. They pointed out that
CPU-GPU systems are the most important system for heterogeneous computing and
matching algorithmic requirements to features of PUs is very important to the
performance.

Based on earlier work [4], this work is focused on leveraging the total
computational power of the CPU and the GPU to achieve high performance for a
finite volume Computational Fluid Dynamics (CFD) solver on CPU-GPU
heterogeneous systems. CFD usually involves compute-intensive computations and
exhibits the feature of single instruction multiple data (SIMD) or single
instruction multiple threads (SIMT) since similar calculations are executed
for a large number of degrees of freedom (finite cells, finite elements,
nodes, etc.). A conventional way of CPU-GPU heterogeneous computing is
offloading the compute-intensive portion of CFD computation to the GPU while
keeping the rest of CFD work on the CPU [5], which is one important reason why
heterogeneous computing system is developed: assigning appropriate
computational tasks to the most suitable components. Different from the
conventional way, this work treats both the CPU and the GPU as suitable
compute components for the residual calculation in the CFD code, assigning
appropriate amounts of workloads to both the CPU and the GPU workers according
to their own speeds. The initiative is to prevent the CPU from idling in the
conventional mode, making full use of the total computational resources in a
single computing system.

Full heterogeneous implementations which utilize modern CPUs and GPUs as equal
compute units are relatively rare. Also, there are no appropriate performance
models to fairly evaluate the performance of CPU-GPU heterogeneous computing.
Alvarez et al. [6] presented a hierarchical heterogeneous implementation
focusing on three algebra-based kernels on hybrid supercomputers. In their
approach, the workload can be distributed among CPUs and GPUs, through the use
of CPU computing threads, GPU management threads, CPU management threads and
CPU communication threads. CPU computing threads are properly bound to NUMA
nodes. Multi-level overlap of communication and computation is shown to be
very useful for computing patterns requiring halo exchanges. However, their
heterogeneous computing approach has not been proven to be effective for a
complete CFD simulation which is usually more complicated. Wang et al. [7]
implemented an “MPI+OpenMP+offload” heterogeneous framework for large-scale
CFD flow field simulations on a heterogeneous CPU-MIC system, with lots of
CPUs and MIC coprocessors being employed. In their framework, each process can
offload some share of its work to the corresponding MIC devices and the
authors scaled their work to half of the Tianhe-2 supercomputer system.
However, the optimizations proposed in their work are tailored to the CPU+MIC
architectures (hardware-specific), limiting its direct applicability to other
heterogeneous computing systems including the CPU+GPU platform. Although the
roofline performance model [8] can be used as guidance on how to improve the
performance of parallel software and hardware by overcoming multiple layers of
”performance ceiling”, the roofline performance model cannot be used to
estimate the advantages of CPU-GPU heterogeneous computing over pure GPU
computing.

##  2 Finite Volume CFD Solver: SENSEI

SENSEI is an acronym for Structured Euler Navier-Stokes Explicit Implicit
solver. SENSEI was initially developed by Derlaga et al [9], and extended to a
turbulence modeling code base by Jackson et al. [10] and Xue et al. [11].
SENSEI adopts multi-block structured grid to ensure mesh quality for general
CFD applications. The governing equations for Navier-Stokes equations can be
written in a weak form as

$\frac{\partial}{\partial t}\int_{\Omega}\vec{Q}{\rm
d}\Omega+\oint_{\partial\Omega}(\vec{F_{i,n}}-\vec{F_{\nu,n}}){\rm
d}A=\int_{\Omega}\vec{S}{\rm d}\Omega$ (1)

where $\vec{Q}$ is the vector of conserved variables, $\vec{F_{i,n}}$ and
$\vec{F_{\nu,n}}$ are the inviscid and viscous flux normal components (the dot
product of the 2nd order flux tensor and the unit face normal vector),
respectively, given as

$\vec{Q}=\begin{bmatrix}\rho\\\ \rho u\\\ \rho v\\\ \rho w\\\ \rho
e_{t}\end{bmatrix},\,\vec{F_{i,n}}=\begin{bmatrix}\rho V_{n}\\\ \rho
uV_{n}+n_{x}p\\\ \rho vV_{n}+n_{y}p\\\ \rho wV_{n}+n_{z}p\\\ \rho
h_{t}V_{n}\end{bmatrix},\,\vec{F_{\nu,n}}=\begin{bmatrix}0\\\
n_{x}\tau_{xx}+n_{y}\tau_{xy}+n_{z}\tau_{xz}\\\
n_{x}\tau_{yx}+n_{y}\tau_{yy}+n_{z}\tau_{yz}\\\
n_{x}\tau_{zx}+n_{y}\tau_{zy}+n_{z}\tau_{zz}\\\
n_{x}\Theta_{x}+n_{y}\Theta_{y}+n_{z}\Theta_{z}\end{bmatrix}$ (2)

$\vec{S}$ is the source term from either body forces, chemistry source terms,
or the method of manufactured solutions [12]. $\rho$ is the density, $u$, $v$,
$w$ are the Cartesian velocity components, $e_{t}$ is the total energy,
$h_{t}$ is the total enthalpy, $V_{n}=n_{x}u+n_{y}v+n_{z}w$ and the $n_{i}$
terms are the components of the outward-facing unit normal vector. $\tau_{ij}$
are the viscous stress components based on Stokes’s hypothesis. $\Theta_{i}$
represents the heat conduction and work from the viscous stresses.

##  3 Discretization

###  3.1 Finite Volume Discretization

Applying finite-volume discretization, the spatial computational domain
$\Omega$ is partitioned into a number of finite control volumes $\Omega_{i}$,
so that $\Omega=\bigcup_{i=1}^{N_{\nu}}\Omega_{i}$, where $N_{\nu}$ is the
number of finite control volumes and the subscript $i$ is the finite control
volume index. The weak form in Eq. 1 can be rewritten for each finite control
volume as

$\frac{\partial}{\partial t}\int_{\Omega_{i}}\vec{U}{\rm
d}\Omega+\oint_{\partial\Omega_{i}}(\vec{F_{i,n}}-\vec{F_{\nu,n}}){\rm
d}s=\int_{\Omega_{i}}\vec{S}{\rm d}\Omega.$ (3)

Denote the discrete solution of the finite volume method as $\vec{U}_{h}$
which is assumed to be constant in each control volume and approximates the
control volume average of the exact solution. The weak form of the general
conservation law can be obtained by integrating the conservation law over each
of the control volumes and applying the Gauss divergence theorem to the volume
integration of the flux divergence term to obtain a surface integral. With the
discrete steady-state residual given as $\vec{R}_{h}$, the discrete version of
Eq. 3 can be given in a semi-discrete form as

$|\Omega_{i}|\dfrac{\partial}{\partial t}\vec{U}_{h}+\vec{R}_{h}=\vec{0},$ (4)

where $|\Omega_{i}|$ is the volume of $\Omega_{i}$, $\vec{U}_{h}$ is the cell
averaged solution vector, $\vec{R}_{h}$ is the spatial residual vector, which
is given in Eq.5.

$\vec{R}_{h}=\sum_{1}^{f}(\vec{F_{i,n}}-\vec{F_{\nu,n}}){\rm\Delta}s-|\Omega_{j}|\vec{S_{h}},$
(5)

where $f$ is the cell face number ($f=4$ and $f=6$ for 2D and 3D,
respectively), $\Delta s$ is the face area, and $S_{h}$ is the cell averaged
source term vector. The discretized equations in Eq. (5) can be marched in
time using an ODE solver if stability conditions are satisfied.

###  3.2 Temporal Discretization

In this work, second-order explicit Runge-Kutta [13, 14, 15] 2-step and 4-step
temporal schemes are mainly applied. The temporal discretization deals with
the numerical approximation of the $\dfrac{\partial}{\partial t}\vec{U}_{h}$
term in Eq. 4. A conventional form of the Runge-Kutta schemes can be seen in
Eq. 6:

$\displaystyle\begin{split}\vec{U}_{h}^{n+1}=&\vec{U}_{h}^{n}-\Delta
t\sum_{i=1}^{s}b_{i}\vec{R}_{h}^{i},\\\
\vec{U}_{h}^{i}=&\vec{U}_{h}^{n}-\Delta
t\sum_{j=1}^{s}a_{ij}\vec{R}_{h}^{j},\end{split}$ (6)

where $s$ is the number of stages, $b_{i}$ and $a_{ij}$ are constants, which
can be found in Ref. [16]. For explicit Runge-Kutta schemes, all $a_{ij}=0$
when $i\leqslant j$.

##  4 Workflow for the GPU-GPU Heterogeneous Computing in SENSEI

The GPU usually has more lightweight compute cores than the CPU so the compute
throughput of the GPU is higher. Besides, the GPU has higher memory bandwidth
and lower memory latency to its own memory. These advantages make the GPU more
appropriate for compute-intensive computations. However, the CPU can handle
versatile tasks including caching, branch prediction, fast single-threaded
computation, etc. Their own advantages make each of them supplementary
components in modern computing systems. The GPU and the CPU in a system
usually have discrete memories so their own data are stored separately, which
requires data exchanges between the CPU and the GPU.

The workflow for the synchronous heterogeneous computing in SENSEI is depicted
in Fig. 1. The CPU mainly deals with the geometry input, domain decomposition,
general CFD settings, residual print, solution output and a small fraction of
computations. Most computations are done on GPU workers with a small amount of
workload assigned to CPU workers according to their relative speed. Boundary
data exchange can happen between CPU workers or between GPU workers, depending
on whether GPUDirect is applied or not. It should be noted that there is a
global synchronization among all CPU workers and GPU workers after the
boundary enforcement for all boundaries for each iteration substep using RK
time marching schemes.

Figure 1: Workflow for the synchronous heterogeneous computing

##  5 Performance Optimizations using OpenACC Directives

Some general guidance exists in terms of improving the performance of a
program in a heterogeneous system. First, sufficient parallelism should be
explored to increase the parallel speedup, that is, the speedup for the
parallel portion should compensate for the overhead of data transfers and the
parallel setup. Second, the memory or network bandwidth should be improved,
which is mainly determined by the data access patterns, data size and data
exchange frequency. Some important performance optimizations using OpenACC
directives are applied to improve the performance of the CPU-GPU heterogeneous
computing in this work.

#### Coalesced memory access

This performance optimization is to improve the memory throughput of the
boundary condition data on each GPU worker. Although the GPU worker has higher
compute capacity compared to the CPU worker, the CPU worker easily encounters
the low memory bandwidth issue. The optimization is targeted at solving this
issue by converting the non-contiguous data into a temporary contiguous array
in parallel using loop for directives and then updating this temporary array
between hosts and devices using update directives.

#### Ghost cell extrapolation on the GPU

Ghost cell extrapolation was initially executed on the CPU host, which
requires ghost cell exchange between the host and the GPU device. One
principle of getting the GPU to work efficiently is keeping the data on the
GPU as long as possible and avoiding frequent data exchange between the host
and the device. In this optimization, all the ghost cell extrapolation for the
GPU is moved to the GPU, instead of execution on the CPU.

#### Performance Optimization on flux limiter calculation

SENSEI applies MUSCL extrapolation to reconstruct fluxes, which is given in
Eq. 7.

$\displaystyle\vec{Q}_{i+1/2}^{L}=$
$\displaystyle\vec{Q}_{i}+\frac{\epsilon}{4}[(1-\kappa)\Psi_{i-1/2}^{+}(\vec{Q}_{i}-\vec{Q}_{i-1})+(1+\kappa)\Psi_{i+1/2}^{-}(\vec{Q}_{i+1}-\vec{Q}_{i})]$
(7) $\displaystyle\vec{Q}_{i+1/2}^{R}=$
$\displaystyle\vec{Q}_{i+1}-\frac{\epsilon}{4}[(1+\kappa)\Psi_{i+1/2}^{+}(\vec{Q}_{i+1}-\vec{Q}_{i})+(1-\kappa)\Psi_{i+3/2}^{-}(\vec{Q}_{i+2}-\vec{Q}_{i+1})]$
(8)

where $\epsilon$ and $\kappa$ are MUSCL extrapolation parameters, $\Psi$ are
limiter function values. $L$ and $R$ denote the left and right states,
respectively.

In the unoptimized version, the limiter variable $\Psi$ is set to be private
and local to every CUDA thread, i.e., different CUDA threads have their own
copies of limiters on the left and right faces to avoid thread contention.
However, this requires the limiter to be computed 2 times redundantly compared
to the limiter calculation on the CPU. To fix this issue, limiters are stored
in a global array and the calculation of limiters is separated from the flux
reconstruction.

##  6 A Performance Model for CPU-GPU Heterogeneous Computing

In this work, a performance model for CPU-GPU heterogeneous computing is
proposed to estimate and predict the performance utilizing both the CPU and
GPU as workers.

For CFD computations using SENSEI or similar CFD code bases, important
computational procedures can be abstracted into high-level computation and
communication patterns and chronologically organized into a workflow chart,
shown in Fig. 2. One single iteration of computation is divided into one
interior domain residual calculation, followed by the boundary condition
enforcement and some data transfers including device-to-host data transfer,
host-to-host data transfer and host-to-device data transfer. Therefore, a
single iteration of the residual calculation can be broken into 5 main
components, each of which is followed by one synchronization to guarantee the
former task is finished. It should be noted that the overhead for these
different synchronizations may not be the same for a real application, but
their effects can be aggregated into a total variable, such as a sequential
fraction.

Figure 2: An abstraction of CPU-GPU heterogeneous computing in SENSEI

For a cubic domain with $N_{l}\times N_{w}\times N_{d}$ interior cells, the
sequential execution time to finish one iteration step on a single CPU is
given in Eq. 9.

$time_{seq}=N_{l}N_{w}N_{d}t_{I}+(2N_{l}N_{w}+2N_{l}N_{d}+2N_{w}N_{d})t_{B}+t_{DH}+t_{HH}+t_{HD}+5t_{S}$
(9)

where $t_{I}$ is the execution time to finish one iteration for one interior
domain cell on the CPU, $t_{B}$ is the execution time to finish one iteration
for one boundary domain cell, $t_{S}$ is the synchronization time, $t_{DH}$,
$t_{HH}$ and $t_{HD}$ represent the time for device-to-host data transfer,
host-to-host data transfer and host-to-device data transfer, respectively.

For a 2D CFD test case using 1D domain decomposition, $N_{d}=1$. For CPU-GPU
heterogeneous computing, decomposing the domain in multiple dimensions can
easily generate a number of small contacting faces which require frequent data
transfers among neighboring blocks. One simple way to avoid this issue is to
decompose the domain only in the $N_{l}$ dimension. If a number of $C$ CPUs
are used to parallelize the residual calculation, the parallel execution time
for the CPU version is given in Eq. 10. Note that no matter how many parallel
CPU cores are used, each decomposed block has a fixed amount of ghost cells in
the decomposition dimension.

$time_{gpu}=\frac{N_{l}N_{w}}{C}t_{I}+(2N_{w}+2\frac{N_{l}}{C})t_{B}+t_{DH}+t_{HH}+t_{HD}+5t_{S}$
(10)

For a scenario in which a number of $G$ GPUs are used, and suppose that the
compute capability of a single GPU is $r_{gc}$ times stronger than a single
CPU, the parallel execution time on multiple GPUs can be derived in Eq. 11.

$time_{gpu}=\frac{N_{l}N_{w}}{Gr_{gc}}t_{I}+(2N_{w}+2\frac{N_{l}}{Gr_{gc}})t_{B}+t_{DH}+t_{HH}+t_{HD}+5t_{S}$
(11)

Combining Eq. 10 and Eq. 11, the execution time for a whole simulation
applying CPU-GPU heterogeneous computing is given in Eq. 12.

$time_{hete}=\frac{N_{l}N_{w}}{Gr_{gc}+C}t_{I}+(2N_{w}+2\frac{N_{l}}{Gr_{gc}+C})t_{B}+t_{DH}+t_{HH}+t_{HD}+5t_{S}$
(12)

It can be seen that we may need to profile each test case to obtain the data
transfer cost and the synchronization cost, which is difficult and
unrealistic. In this work, the influence of the data transfer and
synchronization are aggregated into a total variable, $t_{f}$, which should
not be a constant. $t_{f}$ is affected by the problem size, number of
processes, data transfer size, data transfer frequency, etc. Considering that
the total data transfer cost is strongly affected by connected boundary
conditions, we make an assumption that $t_{f}=\alpha T_{b}$, where $\alpha$ is
a parameter and $T_{b}$ is the total time spent on the boundary enforcement.
In this way, we can obtain Eq. 13:

$time_{hete}=\frac{N_{l}N_{w}}{Gr_{gc}+C}t_{I}+(2N_{w}+2\frac{N_{l}}{Gr_{gc}+C})t_{B}(1+\alpha)$
(13)

In Eq. 13, there are two different times $t_{I}$ and $t_{B}$, representing the
execution time for interior domain and boundary domain, respectively. For a
second-order finite volume CFD code, the relative ratio of $t_{I}$ over
$t_{B}$ is fixed, so we have $t_{B}=\beta t_{I}$, where $\beta$ is in the
range of 0.2$\sim$0.5. Using a profiling tool, the coefficient $\beta$ can be
accurately obtained.

$time_{hete}=[\frac{N_{l}N_{w}}{Gr_{gc}+C}+(2N_{w}+2\frac{N_{l}}{Gr_{gc}+C})(1+\alpha)\beta]t_{I}$
(14)

##  7 Performance Metrics

To compare the performance of a problem with different sizes on different
computing platforms, the wall clock time per iteration step is converted to
_ssspnt_ , which is scaled size steps per np time and defined in Eq. 15.

$\mathrm{ssspnt}=s\frac{size\times steps}{np\times time}$ (15)

where $s$ is a scaling factor to scale _ssspnt_ to the range of [0,1] on slow
computing platforms, $size$ is the problem size denoting the number of finite
cells, $steps$ is the total iteration steps, $np$ is the number of compute
units and $time$ is the CFD solver wall clock time for $steps$ iterations. $s$
is chosen to be $10^{-6}$ in this work.

It is easy to convert the metric _ssspnt_ to FLOPS if we know the operation
count for each iteration. However, it would be difficult for us to manually
count the total number of operations in complicated code, especially in real-
world applications. _ssspnt_ is a better way for the performance evaluation
compared to speedup or efficiency as _ssspnt_ also takes into account the
problem size so that the performance for different problem sizes can be
compared directly. _ssspnt_ enables us to better understand the relative speed
difference than the metric ”efficiency” in different scenarios with various
problem sizes and platforms. Using _ssspnt_ , it is straightforward to know
whether the scaling performance is super-linear, linear, or sub-linear, which
is shown in Fig. 3(a), as well as know the relative speed in different
scenarios including the problems and platforms, which is shown in Fig. 3(b).

(a) Scaling behaviours using _ssspnt_

(b) _ssspnt_ in different scenarios

Figure 3: Advantages of using _ssspnt_ as the performance metric

Every node of the test platform in this work is equipped with two Intel Xeon
E5-2680v4 (Broadwell) 2.4GHz CPUs, 512 GB memory, and two NVIDIA P100 GPUs. A
maximum of 8 nodes are used. Each NVIDIA P100 GPU is capable of up to 4.7
TeraFLOPS of double-precision performance. The compilers used are PGI 18.1 and
Open MPI 3.0.0. A compiler optimization of -O4 is used.

##  8 Results and Discussion

One CFD test case in this work is the supersonic flow through a simplified 2D
30-degree inlet. The CPU serial solution, multi-CPU parallel solution, GPU
serial solution, multi-GPU parallel solution have been carefully compared and
the relative difference among all the solutions is less than ($10^{-12}$).

The inflow conditions are given in Table 1.

Table 1: Supersonic inflow boundary conditions Mach number | 4.0
---|---
Pressure | 12270 Pa
Temperature | 217 K

The Mach number, streamlines and density contours are given in Fig. 4.

(a) Mach number contour and streamlines

(b) Density contour

Figure 4: Solution of 2D supersonic flow

There are some factors affecting the overall performance of CPU-GPU
heterogeneous computing. First, the optimal single GPU worker over single CPU
worker workload ratio needs to be determined, which can be found through
running a series of cases having different workload ratios. As is known, the
GPU worker compute much faster than the CPU worker, so the optimal workload
ratio denotes the balance point between the CPU worker and the GPU worker. The
optimal workload ratio is related to one single GPU over one single CPU
speedup in their pure modes but is likely to be different, as the CPU worker
needs to handle some computational work as well as some control work in the
heterogeneous computing mode, adding synchronization overhead to the whole
computation. Second, the number of GPU workers and the number of CPU workers
may also affect the overall performance. Although a larger number of CPU
workers seems to offer more computational power, the use of more processes may
increase the communication overhead and synchronization overhead. We did
numerical experiments for the Euler solver and NS solver using different
single GPU over CPU workload ratios and different GPU and CPU combinations in
this work.

Fig. 5 shows the _SSSPNT_ performance comparison between RK 2 and RK 4 schemes
in the SENSEI CFD code. It should be noted that the number of substeps is
included in Eq.15. Schemes having more intermediate steps tend to be slightly
more efficient, possibly because more RK step calculations are more compact
and inner sub-step iteration calculations have smaller synchronization
overhead. Overall, the RK 2 and RK 4 temporal schemes have a similar
operational intensity since the Flops per byte of DRAM traffic is similar in
each substep. The performance difference is negligible so we can choose one RK
scheme (RK 4 in this work) to represent a family of RK schemes.

Figure 5: The _SSSPNT_ performance comparison between RK 2 and RK 4

The influence of the problem size is presented in Fig. 6. Generally, the
_SSSPNT_ performance trends on various problem sizes are similar so that a
global optimal single GPU over CPU workload ratio, around 40, can be obtained.
It is easier to achieve higher _SSSPNT_ on larger problems as the GPU needs to
be saturated to work efficiently. This problem can easily become memory
bounded as small workloads are assigned to CPU workers, causing the number of
processes to increase quickly and increasing the communication overhead.
Simple domain decomposition such as 1D domain decomposition should be used to
avoid complicated communication patterns on modern CPU-GPU heterogeneous
systems in which the GPU usually runs much faster than the CPU.

Figure 6: The influence of the problem size

Fig. 7 shows the performance of the Euler solver using _ssspnt_ in SENSEI. It
should be noted that the variable _np_ in Eq. 15 is chosen to be the GPU
number (not counting the CPU number) in the heterogeneous computing mode.
Different numbers of GPU workers and CPU workers as well as different single
GPU/CPU workload ratios are tested. In Fig. 7(a), different curves denote the
use of different numbers of CPU and GPU combinations. It can be found that the
_ssspnt_ increases quickly when increasing the single GPU over single CPU
workload ratio, which indicates that the workload ratio is quickly balanced
between the CPU worker and the GPU worker. When increasing the workload ratio
further, the performance becomes worse as the CPU worker contribution is
negligible compared to the GPU worker contribution. An optimal single GPU over
single CPU workload ratio can be obtained for each CPU-GPU combination. The
optimal workload ratio range is between 40 and 55, which is close to the
single GPU over a single CPU speedup in the pure mode. In Fig. 7(a), there are
some curves using the same number of GPUs but with different numbers of CPUs,
for example, 1 GPU + 8 CPUs vs 1 GPU + 16 CPUs and 2 GPUs + 16 CPUs vs 2 GPUs
+ 32 CPUs. Adding more CPU workers increases the _ssspnt_ , especially when
using a smaller number of GPUs. This can be explained that the communication
cost increases when the number of processes increases, causing the benefits of
heterogeneous computing to drop. Since the goal of applying heterogeneous
computing is to run simulations faster than the pure computing mode, we can
collect all the optimal data points and plot them together, which is the solid
line with the square symbol in Fig. 7(b). One compute unit for the
heterogeneous mode in Fig. 7(b) denotes 1 GPU + 8 CPUs. As can be seen, the
heterogeneous mode is faster than the pure GPU mode, although the performance
gain is not significant. The heterogeneous computing mode may have worse
scaling since the increased number of processes may incur greater
communication overhead. The solid line with the lower triangle symbol
represents the heterogeneous prediction based on the pure GPU and pure CPU
data. The heterogeneous prediction is matched fairly well with the actual
heterogeneous line, which proves the validity of the performance model for
CPU-GPU heterogeneous computing in this work.

(a) The effect of GPU/CPU worker workload ratio

(b) Performance comparison of the heterogeneous computing and pure CPU/GPU
computing mode

Figure 7: _SSSPNT_ Performance of the Euler solver

Fig. 8 shows the _SSSPNT_ performance of the Navier-Stokes solver in the
SENSEI CFD code. The _SSSPNT_ values are smaller than those using the Euler
solver as the Navier-Stokes solver involves the viscous flux calculation while
the Euler solver does not. Overall, the trend of the workload ratio effect in
Fig. 8(a) and the performance comparison of different modes in Fig. 8(b) are
generally similar to what can be summarized for the Euler solver in Fig. 7.
The performance model overpredicts the heterogeneous performance when the
number of compute units is smaller, indicating that the effect of
synchronization and data transfer cost is underestimated. However, the overall
trend of the heterogeneous prediction is fairly reasonable.

(a) The effect of GPU/CPU worker workload ratio

(b) Performance comparison of the heterogeneous computing and pure CPU/GPU
computing mode

Figure 8: _SSSPNT_ Performance of the NS solver

##  9 Conclusions

In this work, a heterogeneous computing implementation for a finite volume CFD
solver on a CPU-GPU system is presented. Its performance has been compared to
a pure-GPU implementation and displays some performance advantages. The
approach can be easily extended to other scientific computing fields to fully
utilize heterogeneous computing resources.

The heterogeneous computing implementation which utilizes a number of CPUs and
GPUs can outperform the pure-GPU computing implementation by up to about 1.23
times. An efficient heterogeneous computing implementation requires the
workload to be distributed reasonably among all co-execution units to match
their own computing capacities, and it also requires the benefits of combining
the total computational power of heterogeneous units to suppress the easily
increased communication overhead. Future computer architects or application
engineers may need to be aware of this when designing a new computer system or
applying heterogeneous computing to accelerate their own applications.

The performance model for CPU-GPU heterogeneous computing proposed in this
work can be used to estimate CFD applications requiring ghost data exchange.
The performance model can also be extended to other scientific computing
domains or help the design of future-generation heterogeneous systems.

## References

  * top [2023] Top500 list for june 2023, 2023. URL: https://www.top500.org/lists/top500/2023/06/, (last accessed on 05/29/23).
  * gre [2023] Green500 list for june 2023, 2023. URL: https://www.top500.org/lists/green500/2023/06/, (last accessed on 05/29/23).
  * Mittal and Vetter [2015] S. Mittal, J. S. Vetter,  A survey of cpu-gpu heterogeneous computing techniques,  ACM Computing Surveys (CSUR) 47 (2015) 69.
  * Xue and Roy [2020] W. Xue, C. J. Roy,  Heterogeneous Computing of CFD Applications on CPU-GPU Platforms using OpenACC Directives,  in: AIAA Scitech 2020 Forum, Orlando, FL, US, 2020, p. 1046\.
  * Xue et al. [2021] W. Xue, C. W. Jackson, C. J. Roy,  An improved framework of gpu computing for cfd applications on structured grids using openacc,  Journal of Parallel and Distributed Computing 156 (2021) 64–85.
  * Álvarez-Farré et al. [2021] X. Álvarez-Farré, A. Gorobets, F. X. Trias,  A hierarchical parallel implementation for heterogeneous computing. application to algebra-based cfd simulations on hybrid supercomputers,  Computers & Fluids 214 (2021) 104768.
  * Wang et al. [2018] Y.-X. Wang, L.-L. Zhang, W. Liu, X.-H. Cheng, Y. Zhuang, A. T. Chronopoulos,  Performance optimizations for scalable cfd applications on hybrid cpu+ mic heterogeneous computing system with millions of cores,  Computers & Fluids (2018).
  * Williams et al. [2009] S. Williams, A. Waterman, D. Patterson,  Roofline: an insightful visual performance model for multicore architectures,  Communications of the ACM 52 (2009) 65–76.
  * Derlaga et al. [2013] J. M. Derlaga, T. Phillips, C. J. Roy,  SENSEI Computational Fluid Dynamics Code: A Case Study in Modern Fortran Software Development,  in: 21st AIAA Computational Fluid Dynamics Conference, San Diego, CA, US, 2013.
  * Jackson et al. [2019] C. W. Jackson, W. C. Tyson, C. J. Roy,  Turbulence Model Implementation and Verification in the SENSEI CFD Code,  in: AIAA Scitech 2019 Forum, San Diego, CA, 2019, p. 2331\.
  * Xue et al. [2020] W. Xue, H. Wang, C. J. Roy,  Code Verification for 3D Turbulence Modeling in Parallel SENSEI Accelerated with MPI,  in: AIAA Scitech 2020 Forum, Orlando, FL, US, 2020, p. 0347\.
  * Oberkampf and Roy [2010] W. L. Oberkampf, C. J. Roy, Verification and validation in scientific computing, Cambridge University Press, 2010.
  * Ascher et al. [1997] U. M. Ascher, S. J. Ruuth, R. J. Spiteri,  Implicit-explicit runge-kutta methods for time-dependent partial differential equations,  Applied Numerical Mathematics 25 (1997) 151–167.
  * Kennedy and Carpenter [2016] C. A. Kennedy, M. H. Carpenter,  Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review (2016).
  * Jameson et al. [1981] A. Jameson, W. Schmidt, E. Turkel,  Numerical Solution of the Euler Equations by Finite Volume Methods using Runge Kutta Time Stepping Schemes,  in: 14th fluid and plasma dynamics conference, Palo Alto, CA, US, 1981, p. 1259\.
  * Ferracina and Spijker [2008] L. Ferracina, M. Spijker,  Strong stability of singly-diagonally-implicit runge–kutta methods,  Applied Numerical Mathematics 58 (2008) 1675–1686.


# Effect of the generalized uncertainty principle on Galilean and Lorentz
transformations

V. M. Tkachuk
Department for Theoretical Physics,
Ivan Franko National University of Lviv,
12 Drahomanov St., Lviv, UA-79005, Ukraine
e-mail<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Generalized Uncertainty Principle (GUP) was obtained in string theory and
quantum gravity and suggested the existence of a fundamental minimal length
which, as was established, can be obtained within the deformed Heisenberg
algebra. We use the deformed commutation relations or in classical case
(studied in this paper) the deformed Poisson brackets, which are invariant
with respect to the translation in configurational space. We have found
transformations relating coordinates and times of moving and rest frames of
reference in the space with GUP in the first order over parameter of
deformation. For the non-relativistic case we find the deformed Galilean
transformation which is similar to the Lorentz one written for Euclidean space
with signature $(+,+,+,+)$. The role of the speed of light here plays some
velocity $u$ related to the parameter of deformation, which as we estimate is
many order of magnitude larger than the speed of light $u\simeq 1.2\times
10^{22}c$. The coordinates of the rest and moving frames of reference for
relativistic particle in the space with GUP satisfy the Lorentz transformation
with some effective speed of light. We estimate that the relative deviation of
this effective speed of light $\tilde{c}$ from $c$ is ${(\tilde{c}-c)/c}\simeq
3.5\times 10^{-45}$. The influence of GUP on the motion of particle and the
Lorentz transformation in the first order over parameter of deformation is
hidden in $1/c^{2}$ relativistic effects.

##  1 Introduction

The investigations in string theory and quantum gravity (see, e.g., [1, 2, 3])
lead to the Generalized Uncertainty Principle (GUP)

$\displaystyle\Delta X\geq{\hbar\over 2}\left({1\over\Delta P}+\beta\Delta
P\right),$ (1)

from which follows the existence of the fundamental minimal length $\Delta
X_{\rm min}=\hbar\sqrt{\beta}$, which, as it is supposed, is of order of
Planck’s length $l_{p}=\sqrt{\hbar G/c^{3}}\simeq 1.6\times 10^{-35}\rm m$. A
broad recent review on this subject can be found in paper [4]. We would like
also to point out the recent discussion around the question whether we can
measure structures with precision better than the Planck’s length, which can
be found in [5].

It was established that minimal length can be obtained in the frame of small
quadratic modification (deformation) of the Heisenberg algebra [6, 7]

$\displaystyle[X,P]=i\hbar(1+\beta P^{2}).$ (2)

In the classical limit $\hbar\to 0$ the quantum-mechanical commutator for
operators is replaced by the Poisson bracket for corresponding classical
variables

$\displaystyle{1\over i\hbar}[X,P]\to\\{X,P\\},$ (3)

which in the deformed case (2) reads

$\displaystyle\\{X,P\\}=(1+\beta P^{2}).$ (4)

We would like to note that historically the first algebra of that kind in the
relativistic case was proposed by Snyder in 1947 [8]. But only investigations
in string theory and quantum gravity renewed the interest in the studies of
physical properties of classical and quantum systems in spaces with deformed
algebras. The observation that GUP can be obtained from the deformed
Heisenberg algebra opens the possibility to study the influence of minimal
length on properties of physical systems on the quantum level as well as on
the classical one.

Deformed commutation relations bring new difficulties in the quantum mechanics
as well as in the classical one. There are known only a few problems, which
can be solved exactly. Namely, one-dimensional harmonic oscillator with
minimal uncertainty in position [6] and also with minimal uncertainty in
position and momentum [9, 10], $D$-dimensional isotropic harmonic oscillator
[11, 12], three-dimensional Dirac oscillator [13], (1+1)-dimensional Dirac
oscillator within Lorentz-covariant deformed algebra [14], one-dimensional
Coulomb problem [15], the singular inverse square potential with a minimal
length [16, 17], the (2+1) dimensional Dirac equation in a constant magnetic
field in the presence of a minimal length [18]. Three-dimensional Coulomb
problem with the deformed Heisenberg algebra was studied within the
perturbation theory in [19, 20], where it was found that common perturbation
theory does not work for $ns$-levels. In [21, 22, 23] the modified
perturbation theory was proposed, which allows to obtain an explicit
expression for corrections to $ns$-levels for hydrogen atom caused by the
deformation of the Heisenberg algebra. In [24] the scattering problem in the
deformed space with minimal length was studied. The ultra-cold neutrons in
gravitational field with minimal length were considered in [25, 26, 27]. The
influence of minimal length on Lamb’s shift, Landau levels, and tunneling
current in scanning tunneling microscope was studied in [28, 29]. The Casimir
effect in a space with minimal length was examined in [30]. In paper [31] the
effect of noncommutativity and of the existence of a minimal length on the
phase space of cosmological model was investigated. The authors of paper [32]
studied various physical consequences, which follow from the noncommutative
Snyder space-time geometry. The gauge invariancy in space with GUP was
considered in [33]. In paper [34] the GUP and localization of a particle in a
discrete space was studied. Some consequences of the GUP-induced ultraviolet
wave-vector cutoff in one-dimensional quantum mechanics was studied in recent
paper [35]. The classical mechanics in a space with deformed Poisson brackets
was studied in [36, 37, 38]. The composite quantum and classical system
($N$-particle system) in the deformed space with minimal length was studied in
[39, 40].

The study of deviation from standard quantum mechanics as well as from
classical one caused by GUP gives a possibility to estimate the upper bound
for minimal length. The collection of upper boundes for minimal length
obtained form the investigation of different properties of different systems
can be found in recent paper [41]. The authors of this paper propose to use
the gravitational bar detectors to place an upper limit for a possible Planck-
scale modifications on the ground-state energy of an oscillator. In [42] the
authors propose to use the quantum-optical control of the mechanical system to
probe a possible deviation from the quantum commutation relation at the Planck
scale.

Note that deformation of the Heisenberg algebra and in classical case
respectively Poisson brackets bring not only technical difficulties in solving
of corresponding equations but also bring problems of a fundamental nature.
One of them is the violation of the equivalence principle in the space with
minimal length [43]. This is the result of assumption that the parameter of
deformation for macroscopic bodies of different mass is unique. In paper [39]
we showed that the center of mass of a macroscopic body in deformed space is
described by an effective parameter of deformation, which is essentially
smaller than the parameters of deformation for particles constituting the
body. Using the result of [39] for the effective parameter of deformation in
[45] we showed that the equivalence principle in the space with minimal length
can be recovered.

In this paper we study the Galilean and Lorentz transformations in space with
deformed Poisson brackets which correspond to the space with minimal length or
GUP. This paper organized as follows. In section 2 starting from a non-
relativistic Hamiltonian we find the Lagrangian of a particle in the space
with deformed Poisson brackets. In section 3 we study the invariancy of action
with the Lagrangian obtained in section 2 and find the deformed Galilean
transformation for coordinates of a non-relativistic particle in one-
dimensional space with GUP. In section 4 this result is generalized for the
three-dimensional case. The Lorentz transformation for coordinates of
relativistic particle in the space with GUP is studied in section 5. And
finally, in section 6 we conclude the results.

##  2 Hamiltonian and Lagrangian of a particle in deformed space

In this section we find the Lagrangian of a classical particle in space with
minimal length starting from the Hamiltonian formalism. It is commonly
supposed that Hamiltonian in deformed case has the form of Hamiltonian in non-
deformed case where instead of canonical variables of non-deformed phase space
are written variables of deformed phase space. So, the Hamiltonian of a
particle (a macroscopic body which we consider as a point particle) of mass
$m$ in the potential $U(X)$ moving in one-dimensional configurational space
reads

$\displaystyle H={P^{2}\over 2m}+U(X),$ (5)

where $X$ and $P$ satisfy deformed Poisson bracket (4). This Poisson bracket
allows the following coordinate representation

$\displaystyle P={1\over\sqrt{\beta}}\tan({\sqrt{\beta}p}),\ \ X=x,$ (6)

where small variables satisfy canonical Poisson bracket

$\displaystyle\\{x,p\\}=1$ (7)

and represent the non-deformed phase space. The Hamiltonian in this
representation reads

$\displaystyle H={\tan^{2}({\sqrt{\beta}p})\over 2m\beta}+U(x).$ (8)

As we see, the deformation of the Poisson bracket in representation (6) is
equivalent to the deformation of kinetic energy.

We consider the linear approximation over the parameter of deformation
$\beta$. In this approximation the Hamiltonian reads

$\displaystyle H={p^{2}\over 2m}+{1\over 3}{\beta\over m}p^{4}+U(x).$ (9)

This Hamiltonian is similar to the relativistic one written in the first order
over $1/c^{2}$

$\displaystyle H_{r}=mc^{2}\sqrt{1+{p^{2}\over
m^{2}c^{2}}}+U(x)=mc^{2}+{p^{2}\over 2m}-{1\over
8m^{3}c^{2}}p^{4}+U(x)+O(1/c^{4}).$ (10)

Introducing effective velocity

$\displaystyle u^{2}={3\over 8\beta m^{2}}.$ (11)

Hamiltonian (9) in the first order over $\beta$ or $1/u^{2}$ can be obtained
from the following one

$\displaystyle H=-mu^{2}\sqrt{1-{p^{2}\over m^{2}u^{2}}}+mu^{2}+U(x).$ (12)

This suggests that corrections to all properties related with deformations
will be similar to relativistic ones in the first order over $1/c^{2}$ but
with an opposite sign before $1/c^{2}$. In particular it suggests that the
Galilean transformations in the first order over $\beta$ will be similar to
the Lorentz one but with an opposite sign before $1/c^{2}$. Let us show it
subsequently.

Because $x$ and $p$ represent the non-deformed canonical space, the Lagrangian
can be found in the traditional way

$\displaystyle L=\dot{x}p-H(x,p),$ (13)

where $p$ is the function of $x$, $\dot{x}$ and can be found from equation

$\displaystyle\dot{x}={\partial H\over\partial p}={p\over m}+{4\over
3}{\beta\over m}p^{3}.$ (14)

In linear over $\beta$ approximation we find

$\displaystyle p=m\dot{x}\left(1-{4\over 3}\beta m^{2}\dot{x}^{2}\right).$
(15)

Substituting it into (13) we finally find the Lagrangian in the linear
approximation over $\beta$

$\displaystyle L={m\dot{x}^{2}\over 2}-{1\over 3}\beta m^{3}\dot{x}^{4}-U(x).$
(16)

Similarly as Hamiltonian (9) this Lagrangian is very similar to the Lagrangian
of a relativistic particle in first order over $1/c^{2}$, namely

$\displaystyle L_{r}=-mc^{2}\sqrt{1-{\dot{x}^{2}\over
c^{2}}}-U(x)=-mc^{2}+{m\dot{x}^{2}\over 2}+{m\over 8c^{2}}\dot{x}^{4}-U(x).$
(17)

The difference is only in constant $mc^{2}$ and opposite sing in the last
term. Thus, we rewrite Lagrangian (16) as follows

$\displaystyle L=mu^{2}\sqrt{1+{\dot{x}^{2}\over u^{2}}}-mu^{2}-U(x),$ (18)

where the effective velocity $u$ is the same as in (11). Of course, Lagrangian
(18) corresponds to (16) only in the first order over $1/u^{2}$ or $\beta$.
The constant $-mu^{2}$ does not influence the equation of motion and can be
omitted.

##  3 Galilean transformation in deformed space

To establish the Galilean transformation it is enough to consider free
particle with Lagrangian (18), where $U=0$. Omitting constant $-mu^{2}$ the
Lagrangian for free particle in first order over $\beta$ reads

$\displaystyle L=mu^{2}\sqrt{1+{\dot{x}^{2}\over u^{2}}}.$ (19)

So, in the first order over parameter of deformation $\beta$ the action reads

$\displaystyle S=mu^{2}\int_{t_{1}}^{t_{2}}\sqrt{1+{\dot{x}^{2}\over
u^{2}}}dt=mu^{2}\int_{(1)}^{(2)}ds,$ (20)

where

$\displaystyle ds^{2}=u^{2}(dt)^{2}+(dx)^{2}$ (21)

is squared interval in the Euclidean space whereas in relativistic case the
second term has an opposite sign and space is pseudo-Euclidean.

Interval (21) is invariant under rotation in plane ($ut,x$). So, symmetry
transformation reads

$\displaystyle x=x^{\prime}\cos\phi+ut^{\prime}\sin\phi,$ (22) $\displaystyle
ut=-x^{\prime}\sin\phi+ut^{\prime}\cos\phi.$ (23)

The angle $\phi$ is related with the velocity $V$ of motion of the point
$x^{\prime}=0$ with respect to the rest frame of reference

$\displaystyle{V\over u}={x\over ut}=\tan\phi.$ (24)

Then Galilean transformation reads

$\displaystyle x={x^{\prime}+Vt^{\prime}\over\sqrt{1+V^{2}/u^{2}}},\ \
t={t^{\prime}-x^{\prime}V/u^{2}\over\sqrt{1+V^{2}/u^{2}}}.$ (25)

We call it the deformed Galilean transformation. This transformation is very
similar to the Lorenz one. The important difference is that here we have an
opposite sign before $1/u^{2}$ that is the result of positive $\beta$, for
which just a minimal length exists. For negative $\beta$ the minimal length is
zero and according to (11) $1/u^{2}$ must be changed to $-1/u^{2}$. In this
case we have common Lorentz transformations where instead of speed of light
$c$ an effective velocity $u$ appears. Note that in fact this transformations
are correct only in the first order over the parameter of deformation $\beta$,
which is related with $1/u^{2}$ [see (11)]. So, in first order over parameter
of deformation we find

$\displaystyle x=(x^{\prime}+Vt^{\prime})\left(1-{V^{2}\over 2u^{2}}\right),\
\ t=t^{\prime}\left(1-{V^{2}\over 2u^{2}}\right)-x^{\prime}{V\over u^{2}}.$
(26)

In the limit $\beta\to 0$ or according to (11) $u\to\infty$ transformation
(26) recover ordinary Galilean transformation. Here it is interesting to note
that transformation (25) or (26) is one of the possible transformations, which
can be obtained in the frame of the following question asked in Special
Relativity Theory: what the most general transformations of spacetime were
that implemented the relativity principle, without making use of the
requirement of the constancy of the speed of light? For details, see section
“Algebraic and Geometric Structures in Special Relativity” in review [44].

Here it is worth to mention the result of paper [45] where we showed that for
a body of mass $m$ the parameter of deformation reads

$\displaystyle\beta={\gamma^{2}\over m^{2}},$ (27)

where $\gamma$ is the same constant for bodies of different mass. It is
interesting to note that constant $c\gamma$ is dimensional. Stress that that
only the relation (27) as was showed in paper [45] leads to recovering of the
equivalence principle in the deformed case. As a result of (27) we have

$\displaystyle u^{2}={3\over 8\gamma^{2}}.$ (28)

and thus the effective velocity does not depend on mass of a body. It means
that Galilean transformation is the same for coordinates of particles of
different mass as everybody feels it must be.

##  4 Three-dimensional case

The generalization of obtained Galilean transformation on three dimensional
case is straightforward. We consider deformed algebra, which is invariant with
respect to translations in configurational space. Different algebras of this
type can be found in [37] (see also references therein). One of the possible
algebra of this type reads

$\displaystyle[X_{i},P_{j}]=i\hbar\sqrt{1+\beta P^{2}}\left(\delta_{i,j}+\beta
P_{i}P_{j}\right),$ (29) $\displaystyle{}[X_{i},X_{j}]=[P_{i},P_{j}]=0.$ (30)

and can be obtained using the representation

$\displaystyle X_{i}=x_{i},\ \ P_{i}={p_{i}\over\sqrt{1-\beta p^{2}}},$ (31)

where ${\bf x}=(x_{1},x_{2},x_{3})$, ${\bf p}=(p_{1},p_{2},p_{3})$ represent
the coordinates and momentum in non-deformed space with canonical commutation
relations

$\displaystyle[x_{i},p_{j}]=\hbar\delta_{i,j},\ \
[x_{i},x_{j}]=[p_{i},p_{j}]=0.$ (32)

Note that in the momentum representation as follows from (31) $p^{2}<1/\beta$
and as a result there is nonzero minimal uncertainty in position or minimal
length. The algebra given by (29) and (30) is invariant with respect to the
transformation $\bf X=\bf X^{\prime}+\bf a$ and thus is translation-invariant
in configurational space. It means that the space is uniform.

Now we consider the classical limit $\hbar\to 0$. Then the deformed Poisson
brackets corresponding to algebra (29), (30) read

$\displaystyle\\{X_{i},P_{j}\\}=\sqrt{1+\beta P^{2}}\left(\delta_{i,j}+\beta
P_{i}P_{j}\right),$ (33)
$\displaystyle{}\\{X_{i},X_{j}\\}=\\{P_{i},P_{j}\\}=0.$ (34)

The Hamiltonian in representation (31) is the following

$\displaystyle H={1\over 2m}{p^{2}\over 1-\beta p^{2}}+U({\bf x})={p^{2}\over
2m}+{\beta\over 2m}p^{4}+U({\bf x})+O(\beta^{2}),$ (35)

where in our consideration we restrict oneself up to to the first order over
$\beta$.

Similarly as in one-dimensional case we find the Lagrangian corresponding to
Hamiltonian (35). First, we find the relation between the velocity and
momentum of the particle

$\displaystyle\dot{x}_{i}={1\over m}{p_{i}\over(1-\beta
p^{2})^{4}}={p_{i}\over m}(1+2\beta p^{2})+O(\beta^{2})$ (36)

and in first order over $\beta$ we obtain

$\displaystyle p_{i}=m\dot{x}_{i}(1-2\beta\dot{x}^{2}).$ (37)

The Lagrangian in this approximation reads

$\displaystyle L={m\dot{\bf x}^{2}\over 2}-{\beta m^{3}\over 2}\dot{\bf
x}^{4}-U({\bf x}).$ (38)

Similarly as in one-dimensional case this Lagrangian in the first order over
$\beta$ can be written in the form (18) and the action of free particle with
$U=0$ for three dimensional case takes form (20) where

$\displaystyle ds^{2}=u^{2}(dt)^{2}+(dx_{1})^{2}+(dx_{2})^{2}+(dx_{3})^{2},$
(39)

here

$\displaystyle u^{2}={1\over 4\beta m^{2}}={1\over 4\gamma^{2}}.$ (40)

In paper [45] from the suggestion that minimal length for electron is of order
of Planck’s length we estimate $\gamma$. Doing similarly we suggest that for
electron $\hbar\sqrt{\beta}=l_{p}$. Then taking into account relation (27) and
substituting for $m$ the mass of electron we find $c\gamma\simeq 4.2\times
10^{-23}$ that reproduce the result of paper [45]. Using this result we find
that $u\simeq 1.2\times 10^{22}c$ which is many order of magnitude large than
the speed of light.

Thus, when the second frame of reference $(t^{\prime},{\bf x^{\prime}})$ moves
with respect to the first one $(t,{\bf x})$ with velocity $V$ along axis
$x_{1}$ then Galilean transformation of coordinate $x^{\prime}_{1}$ and time
$t^{\prime}$ to $x_{1}$ and time $t$ satisfies (26), other coordinates are not
changed $x_{2}=x^{\prime}_{2},\ \ x_{3}=x^{\prime}_{3}$.

##  5 Lorentz transformation in deformed space

In this section we generalize the above consideration for the relativistic
case. Let us start from the one-dimensional relativistic Hamiltonian for free
particle

$\displaystyle H=mc^{2}\sqrt{1+{P^{2}\over m^{2}c^{2}}},$ (41)

where position and momentum satisfy deformed Poisson bracket (4). Using
representation (6) in the first order over $\beta$ and $1/c^{2}$ this
Hamiltonian reads

$\displaystyle H=m^{2}c^{2}+{p^{2}\over 2m}-\left({1\over
8m^{2}c^{2}}-{\beta\over 3}\right){p^{4}\over m}.$ (42)

Introducing notation

$\displaystyle{1\over 8m^{2}\tilde{c}^{2}}={1\over 8m^{2}c^{2}}-{\beta\over
3}$ (43)

we find that this Hamiltonian can be obtained in first order over
$1/\tilde{c}^{2}$ from the following one

$\displaystyle H=m\tilde{c}^{2}\sqrt{1+{p^{2}\over
m^{2}\tilde{c}^{2}}}-m\tilde{c}^{2}+mc^{2}.$ (44)

We suppose that $\beta$ is much smaller than $1/m^{2}c^{2}$. Then this
Hamiltonian corresponds to the relativistic one but with an effective velocity
$\tilde{c}$, which is defined by (43). Note that $\tilde{c}>c$ and
$\tilde{c}\to c$ when $\beta\to 0$. Thus transformation relating coordinates
and time of two reference frames is the Lorentz transformation which contains
instead of speed of light $c$ the effective speed $\tilde{c}$.

Taking into account (27) we find that (43) reads

$\displaystyle{1\over\tilde{c}^{2}}={1\over c^{2}}-{8\over 3}\gamma^{2}$ (45)

and thus the effective speed of light does not depend on the mass of a body.
It means that the Lorentz transformation is the same for particles of
different mass as it must be.

The generalization on three-dimensional case is straitforward. For the
deformed algebra given by (29), (30) we obtain the Hamiltonian in form (44)
where effective velocity is defined by

$\displaystyle{1\over\tilde{c}^{2}}={1\over c^{2}}-4\gamma^{2}.$ (46)

So, similarly as in the one-dimensional case the Lorentz transformation
contains instead of speed of light the effective speed of light. Note that for
different deformed algebras we obtain the same result, only the factor before
$\gamma^{2}$ will be different. In general we can write

$\displaystyle{1\over\tilde{c}^{2}}={1\over c^{2}}-{1\over u^{2}},$ (47)

where $u=\alpha c/\gamma$ and $\alpha$ is a multiplier different for different
algebras. The relative deviation of the effective speed of light $\tilde{c}$
from $c$ in the first order over the parameter of deformation $\beta$ or
$\gamma$ reads

$\displaystyle{\tilde{c}-c\over c}=2c^{2}\gamma^{2}\simeq 3.5\times 10^{-45},$
(48)

here we use that $c\gamma\simeq 4.2\times 10^{-23}$ [see explanation after eq.
(40)].

##  6 Conclusions

In the present paper we have found the transformations relating coordinates
and times of particle in moving and rest frames of reference in the space with
GUP or minimal length in the first order over the parameter of deformation.
For the description of the space with GUP we used the deformed algebra which
is invariant with respect to translation in the configurational space. In the
classical case considered in this paper we have corresponding deformed Poisson
brackets.

For the non-relativistic case we find that this transformation is similar to
the Lorentz one but for space with signature $(+,+,+,+)$. We call it the
deformed Galilean transformation and it is rotation in Euclidian space. The
role of the speed of light here plays some velocity $u$ which is inverse to
${\sqrt{\beta}}m$. It is important to note that, as we shown in our previous
paper [45], the equivalence principle and independence of kinetic energy on
composition of a body require that ${\sqrt{\beta}}m=\gamma$ is constant and
does not depend on the mass of the body. Doing similarly as in paper [45] we
suggest that minimal length for electron is of order of Planck’s length and
set $\hbar\sqrt{\beta}=l_{p}$, then $c\gamma\simeq 4.2\times 10^{-23}$.
Applying this result to the deformed Galilean transformation we find that this
transformation is the same for bodies of different mass as everybody feels it
must be and also estimate the effective velocity which is many orders of
magnitude larger than the speed of light $u\simeq 1.2\times 10^{22}c$.
Therefore, the effect of GUP on the motion of particle and Galilean
transformation is much order smaller the relativistic one. Note that the
deformed Galilean transformation in contrary to ordinal one contains also the
transformation of time which thus is not absolute in space with GUP. In the
limit $\beta\to 0$ or $\gamma\to 0$ the deformed Galilean transformation
recovers the ordinary one. Let us now explain qualitatively why the deformed
Galilean transformation is similar to the Lorentz one. Considering non-
relativistic case we start from the common non-relativistic Hamiltonian
written in deformed variables. But using the representation of deformed
variables over non-deformed ones we find that the Hamiltonian in the first
order over the parameter of deformation contains an additional term
proportional to $p^{4}$. This Hamiltonian is similar to the relativistic one
written in the first order over $1/c^{2}$ but with an opposite sign before
$p^{4}$. This very sign leads to a four-dimensional Euclidean space with
signature $(+,+,+,+)$ in contrary to the ordinary relativistic case with
pseudo-Euclidean space with signature $(+,-,-,-)$. It is interesting to note
that deformed Galilean transformation obtained here for space with GUP is
Euclidian rotation and it is one of the possible transformations, which can be
obtained in the frame of the following question asked in Special Relativity
Theory: what the most general transformations of spacetime were that
implemented the relativity principle, without making use of the requirement of
the constancy of the speed of light? For details, see section “Algebraic and
Geometric Structures in Special Relativity” in review [44].

The similarity of the deformed Galilean transformation to the Lorentz one
forced us to study the relativistic particle in a space with GUP predicting
that the effect of GUP can be hidden in the relativistic effect. We describe
the relativistic particle in space with minimal length or GUP by the
relativistic Hamiltonian which contains deformed variables instead of non-
deformed ones. Using the representation of deformed variables over non-
deformed ones we find that the Hamiltonian in the first order over the
parameter of deformation and first order over $1/c^{2}$ has also a
relativistic form in non-deformed variables with some effective speed of light
$\tilde{c}$. Therefore, coordinates of a relativistic particle in the rest and
moving frames of reference in space with minimal length satisfy the Lorentz
transformation with an effective speed of light. Similarly as in the non-
relativistic case the effective speed of light does not contain the mass of
particle when condition ${\sqrt{\beta}}m=\gamma$ holds and thus in this case
the Lorentz transformation is the same for coordinates and time of particles
of different masses. We estimate that the relative deviation of effective
speed of light $\tilde{c}$ from $c$ is very small ${(\tilde{c}-c)/c}\simeq
3.5\times 10^{-45}$. Finally let us note that the influence of GUP on the
motion of particle and the Lorentz transformation in the first order over the
parameter of deformation is hidden in $1/c^{2}$ relativistic effects.

## Acknowledgment

I am grateful to Dr. T. Masłowski for drawing my attention to review [44].

## References

  * [1] D. J. Gross and P. F. Mende, Nucl. Phys. B 303, 407 (1988).
  * [2] M. Maggiore, 1993 Phys. Lett. B 304, 65 (1993).
  * [3] E. Witten, Phys. Today 49, 24 (1996).
  * [4] S. Hossenfelder, Living. Rev. Relativity 16, 2 (2013).
  * [5] S. Hossenfelder, Class. Quant. Grav. 29. 115011 (2012).
  * [6] A. Kempf, G. Mangano, R. B. Mann, Phys. Rev. D 52, 1108 (1995).
  * [7] A. Kempf, Phys. Rev. D 54, 5174 (1996).
  * [8] H. S. Snyder, Phys. Rev. 71, 38 (1947).
  * [9] C. Quesne and V. M. Tkachuk, J. Phys. A 36, 10373 (2003).
  * [10] C. Quesne and V. M. Tkachuk, J. Phys. A 37, 10095 (2004).
  * [11] L. N. Chang, D. Minic, N. Okamura and T. Takeuchi, Phys. Rev. D 65, 125027 (2002).
  * [12] I. Dadić, L. Jonke and S. Meljanac, Phys. Rev. D 67, 087701 (2003).
  * [13] C. Quesne and V. M. Tkachuk, J. Phys. A 38, 1747 (2005).
  * [14] C. Quesne and V. M. Tkachuk, J. Phys. A 39, 10909 (2006).
  * [15] T. V. Fityo, I. O. Vakarchuk and V. M. Tkachuk, J. Phys. A 39, 2143 (2006).
  * [16] D. Bouaziz, M. Bawin, Phys. Rev. A 76, 032112 (2007).
  * [17] D. Bouaziz, M. Bawin, Phys. Rev. A 78, 032110 (2008).
  * [18] L. Menculini, O. Panella, P. Roy, Phys. Rev. D 87, 065017 (2013).
  * [19] F. Brau, J. Phys. A 32, 7691 (1999).
  * [20] S. Benczik, L. N. Chang, D. Minic and T. Takeuchi, Phys. Rev. A 72, 012104 (2005).
  * [21] M. M. Stetsko and V. M. Tkachuk, Phys. Rev. A 74, 012101 (2006).
  * [22] M. M. Stetsko, Phys. Rev. A 74, 062105 (2006).
  * [23] M. M. Stetsko and V. M. Tkachuk, Phys. Lett. A 372, 5126 (2008).
  * [24] M.M. Stetsko, V.M. Tkachuk, Phys. Rev. A 76, 012707 (2007).
  * [25] F. Brau, F. Buisseret, Phys.Rev.D 74, 036002 (2006).
  * [26] K. Nozari, P. Pedram, EPL 92, 50013 (2010).
  * [27] P. Pedram, K. Nozari, S. H. Taheri, JHEP 1103:093, (2011).
  * [28] S. Das, E. C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008).
  * [29] A. F. Ali, S. Das, E. C. Vagenas, Phys. Rev. D 84, 044013 (2011).
  * [30] A. M. Frassino, O. Panella, Phys. Rev. D 85, 045030 (2012)
  * [31] B. Vakili, Phys. Rev. D 77, 044023 (2008).
  * [32] M. V. Battisti, S. Meljanac, Phys. Rev. D 79, 067505 (2009).
  * [33] M. Kober, Phys.L Rev. D 82, 085017 (2010).
  * [34] M. Bojowald, A. Kempf, Phys. Rev. D 86, 085017 (2012).
  * [35] K. Sailer, Z. Péli, S. Nagy, Phys. Rev. D 87, 084056 (2013).
  * [36] S. Benczik, L. N. Chang, D. Minic, N. Okamura, S. Rayyan, T. Takeuchi, Phys. Rev. D 66, 026003 (2002).
  * [37] A. M. Frydryszak, V. M. Tkachuk, Czechoslovak Journal of Physics 53, No. 11, 5556 C (2003).
  * [38] Z. K. Silagadze, Phys. Lett. A 373, 2643 (2009).
  * [39] C. Quesne, V.M. Tkachuk, Phys. Rev. A 81, 012106 (2010).
  * [40] F. Buisseret, Phys. Rev. A 82, 062102 (2010).
  * [41] F. Marin, F. Marino, M. Bonaldi, et al., Nature Physics, 9, 71 (2012).
  * [42] I. Pikovski, M. R. Vanner, M. Aspelmeyer, at al., Nature Physics 8, 393 (2012).
  * [43] A. F. Ali, Class. Quant. Grav. 28, 065013 (2011).
  * [44] J. Ehlers and C. Lämmerzahl (eds.), Special Relativity [Lect. Notes Phys. 702] (Springer, Berlin–Heidelberg, 2006).
  * [45] V. M. Tkachuk, Phys. Rev. A 86, 062112 (2012).