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Partially explicit splitting scheme with explicit-implicit-null method for nonlinear multiscale flow problems

Yating Wang, Wing Tat Leung

TL;DR

This work incorporates the explicit-implicit-null (EIN) method to separate the nonlinear term into a linear term and a damping term, and utilise the implicit and explicit time marching scheme for the two parts respectively to solve nonlinear high-contrast multiscale diffusion problems.

Abstract

In this work, we present an efficient approach to solve nonlinear high-contrast multiscale diffusion problems. We incorporate the explicit-implicit-null (EIN) method to separate the nonlinear term into a linear term and a damping term, and then utilise the implicit and explicit time marching scheme for the two parts respectively. Due to the multiscale property of the linear part, we further introduce a temporal partially explicit splitting scheme and construct suitable multiscale subspaces to speed up the computation. The approximated solution is splitted into these subspaces associated with different physics. The temporal splitting scheme employs implicit discretization in the subspace with small dimension that representing the high-contrast property and uses explicit discretization for the other subspace. We exploit the stability of the proposed scheme and give the condition for the choice of the linear diffusion coefficient. The convergence of the proposed method is provided. Several numerical tests are performed to show the efficiency and accuracy of the proposed approach.

Partially explicit splitting scheme with explicit-implicit-null method for nonlinear multiscale flow problems

TL;DR

This work incorporates the explicit-implicit-null (EIN) method to separate the nonlinear term into a linear term and a damping term, and utilise the implicit and explicit time marching scheme for the two parts respectively to solve nonlinear high-contrast multiscale diffusion problems.

Abstract

In this work, we present an efficient approach to solve nonlinear high-contrast multiscale diffusion problems. We incorporate the explicit-implicit-null (EIN) method to separate the nonlinear term into a linear term and a damping term, and then utilise the implicit and explicit time marching scheme for the two parts respectively. Due to the multiscale property of the linear part, we further introduce a temporal partially explicit splitting scheme and construct suitable multiscale subspaces to speed up the computation. The approximated solution is splitted into these subspaces associated with different physics. The temporal splitting scheme employs implicit discretization in the subspace with small dimension that representing the high-contrast property and uses explicit discretization for the other subspace. We exploit the stability of the proposed scheme and give the condition for the choice of the linear diffusion coefficient. The convergence of the proposed method is provided. Several numerical tests are performed to show the efficiency and accuracy of the proposed approach.
Paper Structure (17 sections, 5 theorems, 109 equations, 6 figures, 4 tables)

This paper contains 17 sections, 5 theorems, 109 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Fine and coarse scale approximation
  4. Explicit-Implicit-Null (EIN) approach
  5. Partially explicit splitting scheme with EIN
  6. Approximation with proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM)
  7. Construction of multiscale spaces
  8. Construction of basis based on NLMC
  9. Construction of basis based on enriched NLMC (ENLMC)
  10. Stability and convergence
  11. Stability
  12. Convergence
  13. Numerical examples
  14. Example 1
  15. Example 2
  16. ...and 2 more sections

Key Result

Lemma 1

Suppose $C_{1}:=\min_{1\leq n\leq N}\cfrac{\kappa_{u}(u_{H}^{n})-\kappa_{u}(\tilde{u})}{\kappa_{u}(\tilde{u})}<1$, $\|\kappa_{x}^{\frac{1}{2}}\nabla u_{H}^{n+1}\|_{L^{\infty}}<C_{0}$ for all $n$ and $\sup_{v_{2}\in V_{H,2}\backslash\{0\}}\cfrac{\|v_{2}\|_{\kappa}^{2}}{\|v_{2}\|^{2}}\leq\cfrac{(1-\ga Moreover,

Figures (6)

  • Figure 1: Example 1, left: the channelized permeability field, right: the source term.
  • Figure 2: Example 1, the comparison of solutions at different time steps. Left: reference solutions, right: solutions obtained from the proposed algorithm with NLMC basis.
  • Figure 3: Example 1, error history between the reference solution and the approximation solutions.
  • Figure 4: Example 2, left: the channelized permeability field, right: the source term.
  • Figure 5: Example 2, the comparison of solutions at different time steps. Left: reference solutions, right: solutions obtained from the proposed algorithm.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • Theorem 4.1
  • proof
  • Lemma 4
  • proof