Overlapping localized exponential time differencing methods for diffusion problems

Abstract

The paper is concerned with overlapping domain decomposition and exponential time differencing for the diffusion equation discretized in space by cell-centered finite differences. Two localized exponential time differencing methods are proposed to solve the fully discrete problem: the first method is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second method is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergence of the associated iterative solutions to the corresponding fully discrete multidomain solution and to the exact semi-discrete solution is rigorously proved. Numerical experiments are carried out to confirm theoretical results and to compare the performance of the two methods.

Localized Exponential Time DifferencingMethod for Shallow Water Equations: Algorithms and Numerical Study

“…though the difference is very small and in the order of time truncation errors. Note that the same behavior has been observed for diffusion equations in [22,24]). However, as we shall numerically verify in Section 5, the LETD method has the same accuracy as the global ETD method, and it inherits all desirable conservation properties of the TRiSK scheme, namely, conservation of mass, total energy and PV for long time horizons.…”

“…Numerical experiments on three-dimensional coarsening dynamics demonstrated great computational efficiency and excellent parallel scalability of this approach on supercomputers. The overlapping localized ETD was analyzed in [22] and in [23] for the time-dependent diffusion and semi-linear parabolic equations respectively, in which the convergence of the iterative solutions to fully discrete localized ETD solutions and to the exact semi-discrete solution was rigorously proved. A non-overlapping localized ETD was proposed and analyzed for diffusion problems in [24], where the convergence and exact mass conservation were demonstrated.…”

Overlapping domain decomposition based exponential time differencing methods for semilinear parabolic equations

“…To our knowledge, the first literature on numerical analysis of localized ETD algorithms with overlapping domain decomposition was provided by [14] for the diffusion equation in one-dimensional space. For the continuous and space-discrete problems of the diffusion problem, the equivalence of the multidomain problem to the corresponding monodomain one was proven in [11] by showing the convergence of the iterative solutions generated by the Schwarz waveform relaxation algorithm.…”

“…In the fully discrete version, however, the localized ETD scheme is not equivalent to the corresponding monodomain ETD scheme. In [14], the fully discrete first-and second-order localized ETD solutions were proven to converge to the exact solution of the space-discrete multidomain problem. Then, two types of iterative algorithms were proposed for practical calculations: one is based on the Schwarz iteration conducted at each time step and involves solving stationary problems in the subdomains in each iteration, while the other is based on the Schwarz waveform relaxation algorithm where the space-discrete problem is solved in the subdomains in each iteration.…”

“…The iterative solutions were then proven to converge to the fully discrete localized ETD solutions at the same rate as the Schwarz iteration algorithm studied in [11] for the continuous and spacediscrete problems. The analysis given in [14] is mainly based on the maximum principle of the diffusion equation and the corresponding discrete versions. The methods have also been extended to the case of nonoverlapping subdomains in [15], where the convergence of the localized ETD solutions is proven based on the variation-of-constants formula.…”