Accurate and stable grid interfaces for finite volume methods

Abstract

A convection-diffusion equation is discretized by a finite volume method in two space dimensions. The grid is partitioned into blocks with jumps in the grid size at the block interfaces. Interpolation in the cells adjacent to the interfaces is necessary to be able to apply the difference stencils. Second order accuracy is achieved and the stability of the discretizations is investigated. The interface treatment is tested in the solution of the compressible Navier-Stokes equations. The conclusions from the scalar equation are valid also for these equations.

Nondissipative and energy-stable high-order finite-difference interface schemes for 2-D patch-refined grids

“…For two-dimensional meshes, interface closures have been constructed by enforcing Lax stability in a linear scalar advection equation model [7,8]. This approach suffers from dissipation introduced at the interface to stabilize the closure and by the fact that stability is ensured in general only for waves traveling along specific directions.…”

Conservative solution of the Fokker–Planck equation for stochastic chemical reactions

“…The FP equation (2.7) is discretized by a finite volume method in 2D with quadrilateral cells as in [8]. Integrate (2.7) using Gauss' theorem over one cell ω ij in R 2 + to obtain…”

“…The prolongation from a coarse cell to the corresponding four ghost cells in the fine grid has to be fourth order accurate for second order accuracy in the approximation of the second derivatives [8]. In Figure 3.1(a), the upper coarse grid block shares a block boundary with the lower fine grid block.…”

“…They are updated by one-sided interpolation from the coarse cells. One-sided interpolation at the block interfaces and upwind discretization are sufficient for stability for a model problem in [8].…”

A class of energy stable, high-order finite-difference interface schemes suitable for adaptive mesh refinement of hyperbolic problems

“…This has been utilized in second-order finite-volume (FV) [7,8] and finite-difference (FD) [9] methods. The stability properties of the interface stencils are analyzed typically by an eigenvalue analysis, where the rigorous GKS theory [10] is generally a starting point [11,12].…”