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

Shock-resolved Navier–Stokes simulation of the Richtmyer–Meshkov instability start-up at a light–heavy interface

“…Time integration uses the third-order Runge-Kutta (RK32) scheme from Butcher (2003), with uniform time stepping. This approach can be shown to be numerically stable and contribute no numerical dissipation (Kramer et al 2007). The lack of artificial dissipation in the numerical method ensures that only physical viscous effects are observed in the simulation results.…”

“…The recent work of Yee & Sjögreen (2007) uses a shock-capturing sixth-order filter method in their Navier-Stokes (NS) simulation, and as far as the authors are aware, no attempt has been made to resolve the full NS shock structure in the context of an RM simulation until this work. Resolution of the shock and thin contact region is achieved here by multiple levels of local refinement using the stable high-order grid-interface closures developed in Kramer, Pantano & Pullin (2007).…”

Second-order domain decomposition method for three-dimensional hyperbolic problems

“…We compare the proposed SIP scheme of 10 subdomains, SIP(10), with the fully explicit scheme (FES) using Equation (4) and the fully implicit scheme (FIS) using Equation (5 Second, we analyze the efficiency of the proposed SIP scheme. One measurement for the efficiency of a parallel algorithm is parallel CPU time (PCPU) which is defined by total CPU time (TCPU) divided by the number of subdomains (P ).…”

“…There are many implementations on hyperbolic problems such as: finite volume method with grid refinement technique [2], finite difference scheme for adaptive mesh refinement [5], two-time level alternating direction implicit (ADI) finite volume scheme [9], and many others. For three-dimensional hyperbolic problems, there are Galerkin alternating-direction method [6], unconditionally stable ADI method [7], operator splitting technique [8].…”

Stable Grid Refinement and Singular Source Discretization for Seismic Wave Simulations

“…Energy estimation with grid interfaces for finite difference methods can be done by summation by parts difference operators. The summation parts technique has been used extensively for analyzing interfaces for first order hyperbolic problems, both for sudden changes in grid spacing and for discontinuous coefficients in the equation, see for example [20,14] and the references therein. However these articles are limited to constant coefficient problems in one space dimension, thereby not addressing neither the difficulties associated with variable coefficients, nor the difficulties associated with hanging nodes in several space dimensions.…”