Summation by Parts, Projections, and Stability. II

Abstract

Abstract.In this paper we prove strict stability of high-order finite difference approximations of parabolic and symmetric hyperbolic systems of partial differential equations on bounded, curvilinear domains in two space dimensions. The boundary need not be smooth. We also show how to derive strict stability estimates for inhomogeneous boundary conditions. Strict stability in several space dimensionsIn [2] we proved stability for high-order finite difference approximations of hyperbolic and parabolic systems using certain projections and difference operators satisfying a summation-by-parts rule. In one space dimension we showed how to obtain strict stability. The aim of this paper is to prove strict stability in several space dimensions. Furthermore, it will also be demonstrated how to handle inhomogeneous boundary conditions. We limit ourselves to two space dimensions for convenience.The purpose of strict stability is to ensure the same growth rate of the discrete and analytic solutions. If the analytic problem is defined on a curvilinear domain Q with boundary Y (cf. Fig. 1 on next page), then there must exist a diffeomorphism Ç = Ç(x) of Q, onto the unit square (0, 1) x (0, 1) in order for the finite difference method to be well defined. Consequently, a constantcoefficient problem in the original domain may be transformed to a variablecoefficient problem on the unit square, which may account for a nonphysical growth in the discrete estimate.

A kinetic energy preserving nodal discontinuous Galerkin spectral element method

“…We restrict ourselves to the Gauss-Lobatto variant, as in this case the operators satisfy the summation-by-parts (SBP) property, originally introduced in the finite difference framework [22][23][24][25][26],…”

Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points

“…There are several possible stabilisation methods in the literature, such as minmod-type limiting [CS89,CHS90], artificial viscosity methods [PP06, KWH11, RGÖS18, GNA + 19, OGR19], modal filtering [Van91, HK08, GÖS18, RGÖS18], finite volume sub-cells [HCP12, DZLD14, SM14, MO16], and many more. Yet, in [Gas13, KG14, GWK16b] Gassner, Kopriva, and co-authors have been able to construct a DGSEM which is L 2 -stable for certain linear (variable coefficient) as well as nonlinear conservation laws by utilising skew-symmetric formulations of the conservation law (1) and Summationby-Parts (SBP) operators, which were first used and investigated in finite difference (FD) methods [KS74,Str94,Ols95a,Ols95b,NC99]. It should be stressed that the theoretical stability (in the sense of a provable L 2 -norm inequality) as well as the numerical stability (in the sense of stable interpolation polynomials and quadrature rules) of the DGSEM heavily rely on the usage of Gauss-Lobatto points and quadrature weights, which include the boundary nodes and are more dense there, see [Gas13,KG14].…”

Stable high‐order finite‐difference methods based on non‐uniform grid point distributions

“…The non-linear system of equations (15) has to be solved using a Newton-Raphson method. Since the radius of convergence of this method is relatively small, the solution is obtained through a continuation method in q.…”

“…When uniform grids are used, GKS (Gustafsson, Kreiss, and Sundström) theory [9] provides a sufficient test for stability and a recipe for obtaining stable boundary treatments [10, 11] that do not cause radiation of spurious waves into the computational domain [12]. Another approach, based on the energy method and the boundness of the numerical energy, leads to modified stable finite-difference schemes which are known as summation-by-parts methods (SBP) [13][14][15][16]. When non-uniform grids are used, clustering of nodes close to the boundaries cures the instability problems of high-order numerical methods [17][18][19].…”