Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’ equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations.

A systematic literature review of Burgers’ equation with recent advances

SUMMARY
This work presents our strategy, applied optimizations and results in our effort to exploit the computational capabilities of graphics processing units (GPUs) under the CUDA environment in order to solve the Laplacian PDE. The parallelizable red/black successive over-relaxation (SOR) method was used. Additionally, a program for the CPU was developed as a performance reference. Various performance improvements were achieved by using optimization methods, which proved to provide significant speedup. Memory access patterns prove to be a critical factor in efficient program execution on GPUs and it is, therefore, appropriate to follow data reorganization to achieve the highest feasible memory throughput. The same approach exhibits performance benefits on the CPU version, as well. Eventually, a direct comparison of optimal versions’ performance was realized. A 10 × speedup was measured for the CUDA version on an NVidia GTX480 GPU (NVidia Corp, Sta. Clara, CA, USA), exceeding 142 GB/s bandwidth, over the single threaded CPU version when run on an Intel Core i7 2600K CPU. The results prove that the global memory cache added on recent GPU architectures assist achieving high performance without requiring to employ the special memory types provided by the GPU (i.e. shared, texture or constant memory). Copyright © 2012 John Wiley & Sons, Ltd.

Graphics processing unit acceleration of the red/black SOR method

In this paper we study a parallel form of the SOR method for the numerical solution of the Convection Diffusion equation suitable for GPUs using CUDA. To exploit the parallelism offered by GPUs we consider the fine grain parallelism model. This is achieved by considering the local relaxation version of SOR. More specifically, we use SOR with red-black ordering using two sets of parameters @w"1"i"j and @w"2"i"j for the 5 point stencil. The parameter @w"1"i"j is associated with each red (i+j even) grid point (i,j), whereas the parameter @w"2"i"j is associated with each black (i+j odd) grid point (i,j). The use of a parameter for each grid point avoids the global communication required in the adaptive determination of the best value of @w and also increases the convergence rate of the SOR method (Varga, 1962) [38] and (Young, 1971) [41]. We present our strategy and the results of our effort to exploit the computational capabilities of GPUs under the CUDA environment. Additionally, two parallel CPU programs utilizing manual SSE2 (Streaming SIMD Extensions 2) and AVX (Advanced Vector Extensions) vectorization were developed as performance references. The optimizations applied on the GPU version were also considered for the CPU version. Significant performance improvement was achieved with all three developed GPU kernels differentiated by the degree of recomputations thus affecting the flops per element access ratio.

A comparison of CPU and GPU implementations for solving the Convection Diffusion equation using the local Modified SOR method

In this paper we present a new scheme for parallelization of the Successive Over-Relaxation method for solving the Poisson equation over a 3-D volume. Our new scheme is both simple and effective, outperforming the conventional Red-Black scheme by a factor of 16 on an NVIDIA GeForce GTX 590 GPU, a factor of 11 on an NVIDIA GeForce TITAN Black GPU and a factor of 5 on an Intel Xeon Phi. The speed-up compared to the fully optimised reference implementation running on an Intel Xeon CPU is 16 times on the GTX 590, 22 times on the TITAN and 5 times on the Xeon Phi. We explain the rationale and the implementation in OpenCL and present the performance evaluation results.

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Twinned buffering: A simple and highly effective scheme for parallelization of Successive Over-Relaxation on GPUs and other accelerators

Parabolic partial differential equations are often used to model systems involving heat transfer, acoustics, and electrostatics. The need for more complex models with increasing precision drives greater computational demands from processors. Since solving these types of equations is inherently parallel, GPU computing offers an attractive solution for drastically decreasing time to completion, power usage, and increasing the computation per dollar. However, since GPU computing involves a much different programming paradigm than traditional processors, techniques for optimizing solvers must still be developed. This paper presents several optimization strategies for accelerating solvers using CUDA to implement difference equations and compares their performances to a standard processor. The results demonstrate that different strategies should be used for different GPU cards, such as the C1060 and GTX 480, resulting in up to 197 times and 257 times single-precision and up to 133 and 163 times double-precision speedups respectively.

GPU Acceleration of Solving Parabolic Partial Differential Equations Using Difference Equations

Multi-core platform enters the territory of high performance computing (HPC). Moreover, the NVIDA GT200has 240 cores and performs thousands upon thousands of threads simultaneously. The role of the Graphics Processing Units (GPU)accelerator has become more and more important for scientific computing and computational fluid dynamic (CFD) to obtain result quickly and efficiently. In this paper we presented efficient computational scheme for solving parabolic partial differential equations on Multithreading and GPU accelerator. The purpose of this paper is to obtain the solutions as accurate and high speed-up using implicit scheme is convection-diffusion-reaction(CDR). In this paper we propose to CUDA-CDR solver to quickly solve big problem on GPU accelerator. The NVIDIA CUDATM is an extension of C language and has been implemented as a veteran environment since 2006. Our multithread version illustrates 2.5 times faster than those on a sequential code in Model Problem 1 with the problem size of 400x400. The performance of parallel version via CUDA C is 11 times faster than those on a sequential code in Model Problem 1 with the problem size of 400x400.

Solving Parabolic Problems Using Multithread and GPU

Many-Core system plays a key role on High Performance Computing, HPC, nowadays. This platform shows the big potential on the performance per watt, performance per floor area, cost performance, and so on. This paper presents a finite difference scheme solving the general convection-diffusion-reaction equations adapted for application of Graphics Processing Units (GPU) and multithreading. A two-dimensional nonlinear Burgers' equation was chosen as the test case. The best results that we measured are speed-up ratio of 12 times at mesh size 1026×1026 by using GPU and 20 times at mesh size 514×514 by using full 8 CPU cores when compared with an equivalent single CPU code.

Solving burgers' equation using multithreading and GPU

Rapid Performance of Parabolic Problems using Convection Diffusion Reaction on GPU Accelerator.

Multi-core platforms become ubiquitous nowadays. Even laptops contain multi-core processors now. There are multiple cores in a chip or socket or die. A computing node contains multiple chips. Multi-core platforms are rapidly increasing and the number of cores on these platforms is increasing rapidly too. How to enjoy the benefits of parallel computing on the multi-core platforms plays a key role in High Performance Computing. With the increasing complexity of modern multi-core processors, the problem of distributing a software application across different cores to maximize the utilization of the computing power becomes more and more difficult. Different programming patterns great influence the program performance. We implement different parallel programming paradigms on Himeno Benchmark via hybrid MPI/OpenMP in this paper. Moreover, we will evaluate the performance of those on NCHC GPU Cluster and NCHC ALPS. We establish a Roofline Model for NVIDIA GT200, too. We hope the results can give some useful information to the user of HPC.

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Parallel Benefit on Different Programming Paradigms

In the past few years, the graphics processing units (GPU) has become trend in high performance computing (HPC). The newest Top500 list was showed three supercomputers contain GPU accelerator on Top10 in Nov. 2010. The role of the GPU accelerator has become more and more important for scientific computing and computational fluid dynamic (CFD) to obtain result quickly and efficiently. The GPU has become the world's top driving force behind supercomputer. It has hundreds of processor cores in parallel, large-scale operations can be split and can simultaneously load. In this paper we implemented a parallel CDR (pCDR) library of using CUDA. The pCDR provides an easily high- level interface for GPU programming that greatly enhance developer productivity. It was a set of code for solving a convection diffusion reaction (CDR) scalar transport equation. In this paper, we would evaluate the performance comparison of general purpose processor and GPU accelerator. As a result, the performance of pCDR-CG via CUDA C has 6 times faster than those on a sequential code in the problem size of 800× 800.

Sheng-Hsiu Kuo

Papers

Solving Parabolic Problems Using Multithread and GPU

Solving burgers' equation using multithreading and GPU

Rapid Performance of Parabolic Problems using Convection Diffusion Reaction on GPU Accelerator.

Parallel Benefit on Different Programming Paradigms

Evaluation iterative solver for pCDR on GPU accelerator