2025/04/02 by Engwer, Christian, Schell, Alexander, Dreier, Nils-Arne
#35K61 #49M15 #65J10 #65J15 #65M20 #65Y05 #65Y10 #86-08 #Computational Engineering #FOS: Computer and information sciences #FOS: Mathematics #Finance #Mathematical Software (cs.MS) #Numerical Analysis (math.NA) #and Science (cs.CE)
paper · doi:10.48550/arxiv.2504.02117
High order methods have shown great potential to overcome performance issues of simulations of partial differential equations (PDEs) on modern hardware, still many users stick to low-order, matrixbased simulations, in particular in porous media applications. Heterogeneous coefficients and low regularity of the solution are reasons not to employ high order discretizations. We present a new approach for the simulation of instationary PDEs that allows to partially mitigate the performance problems. By reformulating the original problem we derive a parallel in time time integrator that increases the arithmetic intensity and introduces additional structure into the problem. By this it helps accelerate matrix-based simulations on modern hardware architectures. Based on a system for multiple time steps we will formulate a matrix equation that can be solved using vectorised solvers like Block Krylov methods. The structure of this approach makes it applicable for a wide range of linear and nonlinear problems. In our numerical experiments we present some first results for three different PDEs, a linear convection-diffusion equation, a nonlinear diffusion-reaction equation and a realistic example based on the Richards' equation.