Benchmarking solvers for the one dimensional cubic nonlinear klein gordon equation on a single core

Abstract

To determine the best method for solving a numerical problem modeled by a partial differential equation, one should consider the discretization of the problem, the computational hardware used and the implementation of the software solution. In solving a scientific computing problem, the level of accuracy can also be important, with some numerical methods being efficient for low accuracy simulations, but others more efficient for high accuracy simulations. Very few high performance benchmarking efforts allow the computational scientist to easily measure such tradeoffs in order to obtain an accurate enough numerical solution at a low computational cost. These tradeoffs are examined in the numerical solution of the one dimensional Klein Gordon equation on single cores of an ARM CPU, an AMD x86-64 CPU, two Intel x86-64 CPUs and a NEC SX-ACE vector processor. The work focuses on comparing the speed and accuracy of several high order finite difference spatial discretizations using a conjugate gradient linear solver and a fast Fourier transform based spatial discretization. In addition implementations using second and fourth order timestepping are also included in the comparison. The work uses accuracy-efficiency frontiers to compare the effectiveness of five hardware platforms