ABSTRACT In a recent paper, Slepian and Philcox derive an explicit solution of the homologous collapse from rest uniform density sphere under its self-gravity as function time. Their is given in terms two curvilinear integrals along suitable Jordan contour; practice, it must be approximated by quadrature rule. The aim this paper to examine how choice contour rule affects accuracy efficiency int...

Abstract This article proposes an efficient concurrent coupling of two different material scales—a macroscale and a microscale—in direct solution scheme based on explicit time integration. Both scales may be discretized with element sizes the microdomain exhibit heterogeneous structure. A surface is described, which imposes macrovelocities at interfaces microscale. Using averaged stress state s...

We describe and evaluate various implicit and semiimplicit time integration schemes applied to the numerical simulation of hydrodynamical and magnetohydrodynamical problems. The schemes were implemented recently in the software package Versatile Advection Code, which uses modern shock capturing methods to solve systems of conservation laws with optional source terms. The main advantage of impli...

We study the design of a general, fully explicit numerical method for simulating kinetic equations with an extended BGK collision model allowing for multiple relaxation times. In that case, the problem is stiff and we show that its spectrum consists of multiple separated eigenvalue clusters. Projective integration methods are explicit integration schemes that first take a few small (inner) step...

Specialized integration algorithms for initial value problems, obtained by applying conventional explicit discretizations in a transformed space, are described. One example, conservative integration, is motivated by a theorem of Ge Zhong and Marsden [17] that establishes that in the absence of explicit time dependence, one must in practice choose between preserving symplecticity or conserving t...