Shock capturing by Bernstein polynomials for scalar conservation laws

Abstract

A main disadvantage of many high-order methods for hyperbolic conservation laws lies in the famous Gibbs-Wilbraham phenomenon, once discontinuities appear in the solution. Due to the Gibbs-Wilbraham phenomenon, the numerical approximation will be polluted by spurious oscillations, which produce unphysical numerical solutions and might finally blow up the computation. In this work, we propose a new shock capturing procedure to stabilise high-order spectral element approximations. The procedure consists of going over from the original (polluted) approximation to a convex combination of the original approximation and its Bernstein reconstruction, yielding a stabilised approximation. The coefficient in the convex combination, and therefore the procedure, is steered by a discontinuity sensor and is only activated in troubled elements. Building up on classical Bernstein operators, we are thus able to prove that the resulting Bernstein procedure is total variation diminishing and preserves monotone (shock) profiles. Further, the procedure can be modified to not just preserve but also to enforce certain bounds for the solution, such as positivity. In contrast to other shock capturing methods, e. g. artificial viscosity methods, the new procedure does not reduce the time step or CFL condition and can be easily and efficiently implemented into any existing code. Numerical tests demonstrate that the proposed shock-capturing procedure is able to stabilise and enhance spectral element approximations in the presence of shocks.

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

“…reached) or additional shock-capturing. Shock-capturing might be performed, for instance, by artificial viscosity methods [PP06, KWH11, GNA + 19, RG ÖS18], modal filtering [Van91, HK08, MOSW13, G ÖS18], finite volume subcells [HCP12, SM14, DZLD14, MO16], or other methods[GG19,Gla19]. Shock capturing in DLS based high-order methods might be investigated in future works.…”

The physics of financial networks

“…Furthermore, it appears that the differentiation matrices of RBF methods encountered in time-dependent PDEs often have eigenvalues with a positive real part resulting in unstable methods; see [76]. Hence, in the presence of rounding errors, these methods are less accurate [55,75,80] and can become unstable in time unless a dissipative time integration method [64,76], artificial dissipation [22,77,39,37,73], or some other stabilizing technique [79,28,35,48,40,30,15] is used. So far, this issue was only overcome for problems which are free of BCs [64].…”

“…In the context of RBF methods, appropriate discrete norms would correspond to stable high-order quadrature/cubature rules for (potentially) scattered data points [52,34,33,31]. For nonlinear problems, sometimes also the total variation is considered; see [47,10,11,87,30] and references therein.…”

Towards Stable Radial Basis Function Methods for Linear Advection Problems

“…In the context of RBF methods, appropriate discrete norms would correspond to stable high-order quadrature/cubature rules for (potentially) scattered data points [52,34,33,31]. For nonlinear problems, sometimes also the total variation is considered; see [47,10,11,87,30] and references therein.…”