A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations

Abstract

Abstract. A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear PDEs the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a complete time interval. Nonlinear terms are treated as a source term, provided by the solution from the previous iteration. At each iteration, the problem is decoupled into independent subproblems by the principle of superposition. The decoupled subproblems are solved fast by exponential integration, based on a block Krylov method. The new time integration is demonstrated for the one-dimensional advection-diffusion equation and the viscous Burgers equation. Numerical experiments confirm excellent parallel scaling for the linear advection-diffusion problem, and good scaling in case the nonlinear Burgers equation is simulated.Key words. parallel computing, exponential integrators, partial differential equations, parallel in time, block Krylov subspace.AMS subject classifications. 65F60, 65L05, 65Y051. Introduction. Recent developments in hardware architecture, bringing to practice computers with hundreds of thousands of cores, urge the creation of new, as well as a major revision of existing numerical algorithms [14]. To be efficient on such massively parallel platforms, the algorithms need to employ all possible means to parallelize the computations. When solving partial differential equations (PDEs) with a time-dependent solution, an important way to parallelize the computations is, next to the parallelization across space, parallelization across time. This adds a new dimension of parallelism with which the simulations can be implemented. In this paper we present a new time-parallel integration method extending the Paraexp method [21] to nonlinear partial differential equations using Krylov methods and waveform relaxation.Several approaches to parallelize the simulation of time-dependent solutions in time can be distinguished. The first important class of the methods are the waveform relaxation methods [6,38,44,56], including the space-time multigrid methods for parabolic PDEs [8,24,30,37]. The key idea is to start with an approximation to the numerical solution for the whole time interval of interest and update the solution, solving an easier-to-solve approximate system in time. The Parareal method [36], which attracted significant attention recently, is a prime example related to the class of waveform relaxation methods [19].Parallel Runge-Kutta methods and general linear methods, where the parallelism is determined and restricted by the number of stages or steps, form another class of the time-parallel methods [8,12,52]. Time-parallel schemes can also be obtained

Towards parallel-in-time simulations of turbulent Rayleigh-Bénard convection

“…Exponential integrators are in general attractive due to their excellent stability and accuracy properties [77]. The EBK method is demonstrated to be particularly competitive for solving (5.1) compared to implicit schemes, and also with respect to other exponential integrators [98]. The EBK method is based on a so-called block Krylov subspace, which is generated by the action of a matrix on multiple vectors simultaneously.…”

“…The nonlinearity of the problem can be dealt with by incorporating the EBK method into an iterative procedure [98]. Namely, the nonlinear term is linearized about a stateū, specified later, as follows…”

“…An extension of Paraexp to nonlinear initial-value problems has been introduced in [98]. Preliminary tests show that parallel speedup is realistic for the viscous Burgers equation, even for low viscosity coefficients.…”

“…We find that the EBK method can be applied successfully to incompressible flows, also in cases with rather low viscosity. Furthermore, we show that the EBK method can be used within the time-parallel framework outlined in [98]. We provide a simplified model to analyze how much speedup is feasible with the EBK method for parallel-in-time simulations of the incompressible Navier-Stokes equation.…”

“…Preliminary tests show that parallel speedup is realistic for the viscous Burgers equation, even for low viscosity coefficients. The approach in [98] is based on the exponential block Krylov (EBK) method [11]. A difference with the original formulation of Paraexp is that the exponential integrator, here the EBK method, is used to solve both the nonhomogeneous and the homogeneous subproblems.…”