A novel numerical viscosity for fourth order hybrid entropy stable shock capturing schemes for convection diffusion equation

• Published in: Journal of computational physics 2022-12, Vol.470, p.111586, Article 111586
• Main Authors: Jisha, C.R., Riyasudheen, T.K., Dubey, Ritesh Kumar
• Format: Article
• Language: English
• Subjects: Central difference operator; Convection diffusion equation; Entropy stable schemes; Numerical diffusion
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111586

•Constructed a novel way of numerical viscosity for the fourth-order entropy stable scheme.•The significant advantage of the current work is the simplicity rather than existing numerical viscosity.•It is not necessary to compute the sign stable jump of the fourth-order reconstruction of a scaled variable when applying high-order reconstruction methods such as ENO/WENO/TENO.•Introduced technique is possible to extend arbitrary higher order and multidimensional entropy stable scheme. This work presents a novel numerical viscosity for constructing fourth-order hybrid entropy stable schemes for convection-diffusion equations. The numerical diffusion is designed using the fourth-order central difference approximation of second-order spatial derivative. By construction, the proposed numerical diffusion is efficient compared to the existing approaches, which depend on higher order jump in the scaled entropy variable using computationally expensive sign stable higher order reconstructions and a suitable diffusion matrix. The resulting schemes are shown entropy stable. The constructed schemes are extended to the systems and further modified using a hybrid approach to suppress the oscillations near discontinuities. Numerical results are presented to support the implementation of the proposed schemes.