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Multicontinuum splitting scheme for multiscale flow problems

Yalchin Efendiev, Wing Tat Leung, Buzheng Shan, Min Wang

Abstract

In this paper, we propose multicontinuum splitting schemes for multiscale problems, focusing on a parabolic equation with a high-contrast coefficient. Using the framework of multicontinuum homogenization, we introduce spatially smooth macroscopic variables and decompose the multicontinuum solution space into two components to effectively separate the dynamics at different speeds (or the effects of contrast in high-contrast cases). By treating the component containing fast dynamics (or dependent on the contrast) implicitly and the component containing slow dynamics (or independent of the contrast) explicitly, we construct partially explicit time discretization schemes, which can reduce computational cost. The derived stability conditions are contrast-independent, provided the continua are chosen appropriately. Additionally, we discuss possible methods to obtain an optimized decomposition of the solution space, which relaxes the stability conditions while enhancing computational efficiency. A Rayleigh quotient problem in tensor form is formulated, and simplifications are achieved under certain assumptions. Finally, we present numerical results for various coefficient fields and different continua to validate our proposed approach. It can be observed that the multicontinuum splitting schemes enjoy high accuracy and efficiency.

Multicontinuum splitting scheme for multiscale flow problems

Abstract

In this paper, we propose multicontinuum splitting schemes for multiscale problems, focusing on a parabolic equation with a high-contrast coefficient. Using the framework of multicontinuum homogenization, we introduce spatially smooth macroscopic variables and decompose the multicontinuum solution space into two components to effectively separate the dynamics at different speeds (or the effects of contrast in high-contrast cases). By treating the component containing fast dynamics (or dependent on the contrast) implicitly and the component containing slow dynamics (or independent of the contrast) explicitly, we construct partially explicit time discretization schemes, which can reduce computational cost. The derived stability conditions are contrast-independent, provided the continua are chosen appropriately. Additionally, we discuss possible methods to obtain an optimized decomposition of the solution space, which relaxes the stability conditions while enhancing computational efficiency. A Rayleigh quotient problem in tensor form is formulated, and simplifications are achieved under certain assumptions. Finally, we present numerical results for various coefficient fields and different continua to validate our proposed approach. It can be observed that the multicontinuum splitting schemes enjoy high accuracy and efficiency.
Paper Structure (15 sections, 5 theorems, 68 equations, 17 figures, 7 tables)

This paper contains 15 sections, 5 theorems, 68 equations, 17 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multicontinuum splitting schemes
  4. Optimized decomposition of multicontinuum space
  5. Construction for discretization scheme 1
  6. Construction for discretization scheme 2
  7. Numerical examples
  8. Two-continuum cases
  9. Example 1
  10. Example 2
  11. Multiple-continuum cases
  12. Example 3
  13. Example 4
  14. Example 5
  15. Conclusions

Key Result

Lemma 1

Let $H$ be a Hilbert space and $F_1,F_2 \subset H$ be two finite dimensional subspaces. If $F_1 \cap F_2 = \{0\}$, then there exists a constant $\gamma=\gamma(F_1,F_2)\in [0,1)$ such that for all $x \in F_1$ and $y \in F_2$.

Figures (17)

  • Figure 5.1: Source term $f$
  • Figure 5.2: Left: Layered field $\kappa$ in Example 1. Right: Reference solution at the final time $T$ when $\max \kappa = 10^5$ in Example 1.
  • Figure 5.3: Relative $L^2$ error for different schemes when $H=1/10$ and $l=5$ in Example 1. Left: $e^{(1)}(t)$. Right: $e^{(2)}(t)$.
  • Figure 5.4: Relative $L^2$ error for different schemes when $H=1/20$ and $l=6$ in Example 1. Left: $e^{(1)}(t)$. Right: $e^{(2)}(t)$.
  • Figure 5.5: Relative $L^2$ error for different schemes when $H=1/40$ and $l=8$ in Example 1. Left: $e^{(1)}(t)$. Right: $e^{(2)}(t)$.
  • ...and 12 more figures

Theorems & Definitions (8)

  • Lemma 1: Strengthened Cauchy-Schwarz inequality
  • Theorem 2
  • proof
  • Theorem 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof