Contrast-independent partially explicit time discretizations for multiscale flow problems

Abstract

Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extend are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is dt = H 2 /κ max , where κ max is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The unconditional stability of these algorithms require a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented.

Partially Explicit Time Discretization for Nonlinear Time Fractional Diffusion Equations

“…This condition is similar to that of the nonlinear non-fractional PDE [12]. By careful construction of spaces, the stability condition is contrast-independent [24]. We also find that the restrictive time step scales as the coarse mesh size and thus can be much larger.…”

“…In this section, we present a method to build the two spaces satisfying (3.3). Here, we follow our previous work [24]. We will first present the CEM finite element space which is used as V H,1 .…”

“…Earlier approaches include implicit-explicit approaches and other techniques [4,38,1,23,3,40,47,16,25,34,45,33,15]. In recent works, splitting methods are designed to solve the linear and nonlinear parabolic partial differential equations [24,12]. This paper applies the splitting method to time fractional partial differential equations.…”

Contrast-independent partially explicit time discretizations for nonlinear multiscale problems

“…The main drawback of explicit methods is that they require small time steps that scale as the fine mesh and depend on physical parameters, e.g., the contrast. To alleviate this issue, we propose a novel nonlinear splitting algorithm following our earlier works [23,24] for linear equations. The main idea of our approaches is to use multiscale methods on a coarse spatial grid such that the time step scales as the coarse mesh size.…”

“…Our proposed approaches follow concepts developed in [23,24] for linear equations. In these works, we design splitting algorithms for solving flow and wave equations.…”