A simple shock-capturing technique for high-order discontinuous Galerkin methods

• Published in: International journal for numerical methods in fluids Vol. 69; no. 10; pp. 1614 - 1632
• Main Authors: Huerta, A., Casoni, E., Peraire, J.
• Format: Journal Article Publication
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 10.08.2012; Wiley
• Subjects: Adaptivity; Anàlisi numèrica; Compressible flow; Compressible flows; shock and detonation phenomena; Computational methods in fluid dynamics; Discontinous enrichment; Discontinuous Galerkin; Euler equations; Euler flow; Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Física; Física de fluids; Galerkin methods; Galerkin, Mètodes de; High order; Matemàtiques i estadística; Mètodes en elements finits; Physics; Shock-capturing; Shock-wave interactions and shock effects; Àrees temàtiques de la UPC; Supersonic flow; Compressible fluid; Discontinuous Galerkin method; Transonic flow; Shock-capturing; Computational fluid dynamics; Euler flow; compressible flow; high order; Digital simulation; Convergent divergent nozzle; Euler equations; Step; Adaptive method; discontinous enrichment; Shock waves; Hump; adaptivity; discontinuous Galerkin; Modelling; Mesh generation
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.2654

SUMMARY This article presents a novel shock‐capturing technique for the discontinuous Galerkin (DG) method. The technique is designed for compressible flow problems, which are usually characterized by the presence of strong shocks and discontinuities. The inherent structure of standard DG methods seems to suggest that they are especially adapted to capture shocks because of the numerical fluxes based on suitable approximate Riemann solvers, which, in practice, introduces some stabilization. However, the usual numerical fluxes are not sufficient to stabilize the solution in the presence of shocks for large high‐order elements. Here, a new basis of shape functions is introduced. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization because of numerical fluxes. Large high‐order elements can therefore be used and shocks captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Several numerical examples for transonic and supersonic flows are studied to demonstrate the applicability of the proposed approach. Copyright © 2011 John Wiley & Sons, Ltd.