Michael Quell

S. Aseeri, O. Batrašev, M. Icardi et al.

The cubic Klein-Gordon equation is a simple but non-trivial partial differential equation whose numerical solution has the main building blocks required for the solution of many other partial differential equations. In this study, the library 2DECOMP&FFT is used in a Fourier spectral scheme to solve the Klein-Gordon equation and strong scaling of the code is examined on thirteen different machines for a problem size of 512^3. The results are useful in assessing likely performance of other parallel fast Fourier transform based programs for solving partial differential equations. The problem is chosen to be large enough to solve on a workstation, yet also of interest to solve quickly on a supercomputer, in particular for parametric studies. Unlike other high performance computing benchmarks, for this problem size, the time to solution will not be improved by simply building a bigger supercomputer.

Winfried Auzinger, Othmar Koch, Michael Quell

We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several applications generating intricate patterns and displaying nonsmooth solution dynamics. First we give a general error analysis for splitting methods for parabolic problems under periodic boundary conditions and derive the necessary smoothness requirements on the exact solution in particular for the Gray-Scott equation and the Van der Pol equation. Numerical examples demonstrate the convergence of the methods and serve to compare the efficiency of different time-adaptive splitting schemes and of splitting into either two or three operators, based on appropriately constructed a posteriori local error estimators.