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.

Accepted in Applied Mathematics and Computation

References in corpus (4)

• Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method
• Smooth and compactly supported viscous sub-cell shock capturing for Discontinuous Galerkin methods
• Application of Modal Filtering to a Spectral Difference Method
• High Order Edge Sensors with ${\ell }^1$ Regularization for Enhanced Discontinuous Galerkin Methods

Cited by in corpus (4)

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• Towards Stable Radial Basis Function Methods for Linear Advection Problems
• Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions
• High-Order Curvilinear Finite Element Magneto-Hydrodynamics I: A Conservative Lagrangian Scheme