Abstract
We present a parallel GPU solution of the Caputo fractional reaction-diffusion equation in one spatial dimension with explicit finite difference approximation. The parallel solution, which is implemented with CUDA programming model, consists of three procedures: preprocessing, parallel solver, and postprocessing. The parallel solver involves the parallel tridiagonal matrix vector multiplication, vector-vector addition, and constant vector multiplication. The most time consuming loop of vector-vector addition and constant vector multiplication is optimized and impressive performance improvement is got. The experimental results show that the GPU solution compares well with the exact solution. The optimized GPU solution on NVIDIA Quadro FX 5800 is 2.26 times faster than the optimized parallel CPU solution on multicore Intel Xeon E5540 CPU.
Highlights
The idea of fractional derivatives can be dated back to the 17th century
The domain decomposition algorithm keeps the same parallelism but needs much fewer iterations, compared with Jacobi iteration in each time step, until nothing has been recorded on accelerating the numerical solution of Caputo fractional reaction-diffusion equation on graphics processing unit (GPU)
This paper focuses on the Caputo fractional reactiondiffusion equation: 0Dtαu (x, t) μu (x, t)
Summary
The idea of fractional derivatives can be dated back to the 17th century. A fractional differential equation is a kind of equation which uses fractional derivatives. In 2000, Henry and Wearne [12] derived a fractional reaction-diffusion equation from a continuous-time random walk model with temporal memory and sources. The generalized differential transform method [15] was presented for fractional reaction-diffusion equations. GPU presents an energy efficient architecture for computation intensive domains like particle transport [24, 25] and molecular dynamics [26]. Domain decomposition method is regarded as the basic mathematical background for many parallel applications. A domain decomposition algorithm for time fractional reaction-diffusion equation with implicit finite difference method was proposed [33]. The domain decomposition algorithm keeps the same parallelism but needs much fewer iterations, compared with Jacobi iteration in each time step, until nothing has been recorded on accelerating the numerical solution of Caputo fractional reaction-diffusion equation on GPU.
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