Temporal Splitting Schemes for Multiscale Problems

Abstract

Many real-life problems have a multiscale nature and high contrast coefficients. High contrast is known to pose significant challenges in numerical simulation, especially for time-dependent problems. Implicit methods are commonly used for time discretization; although they are unconditionally stable, they require expensive computations at each time step. In contrast, explicit methods are computationally efficient per time step but require very small time steps to maintain stability, due to the mesh size and contrast. We propose temporal splitting algorithms for multiscale problems in mixed form that balance the efficiency of explicit methods with the stability of implicit methods. This approach is applied to flow, wave, and quasi-gas-dynamic (QGD) problems. We decompose the pressure space into two components: a coarse-grid part and a correction part. Each is paired with a corresponding velocity space, namely a coarse-grid velocity space and a correction velocity space. The coarse-grid subspaces are designed to capture fast-scale features influenced by high-contrast variations, while the correction subspaces account for slow-scale features that are independent of contrast and not resolved by the coarse-grid approximation. Using this decomposition, we develop temporal splitting schemes that treat fast components implicitly and slow components explicitly. The stability of the proposed algorithms is ensured through a careful design of multiscale spaces. We also show that time stepping in the correction spaces is independent of contrast. Additionally, we find that the allowable time step scales with the coarse mesh size, which can lead to significant computational savings. We further propose multicontinuum splitting schemes for the wave equation and the quasi-gas-dynamic problem with high-contrast coefficients. To separate fast and slow dynamics in the system, we decompose the solution space into two components. This is achieved by introducing physically meaningful macroscopic variables and employing the expansion in multicontinuum homogenization. Based on this decomposition, we formulate partially explicit time discretization schemes in which the fast (contrast-dependent) component is treated implicitly to ensure stability, while the slow (contrast-independent) component is treated explicitly to enhance computational efficiency. We introduce the concept of discrete energy and derive corresponding stability conditions, which remain independent of contrast when the continua are properly chosen. In addition, we discuss strategies for optimizing the space decomposition. Numerical examples are presented to validate the accuracy and stability of the proposed schemes.