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We introduce an adaptive viscosity regularization approach for the numericalsolution of systems of nonlinear conservation laws with shock waves. Theapproach seeks to solve a sequence of regularized problems consisting of thesystem of conservation laws and an additional Helmholtz equation for theartificial viscosity. We propose a homotopy continuation of the regularizationparameters to minimize the amount of artificial viscosity subject topositivity-preserving and smoothness constraints on the numerical solution. Theregularization methodology is combined with a mesh adaptation strategy thatidentifies the shock location and generates shock-aligned meshes, which allowsto further reduce the amount of artificial dissipation and capture shocks withincreased accuracy. We use the hybridizable discontinuous Galerkin method tonumerically solve the regularized system of conservation laws and thecontinuous Galerkin method to solve the Helmholtz equation for the artificialviscosity. We show that the approach can produce approximate solutions thatconverge to the exact solution of the Burgers' equation. Finally, wedemonstrate the performance of the method on inviscid transonic, supersonic,hypersonic flows in two dimensions. The approach is found to be accurate,robust and efficient, and yields very sharp yet smooth solutions in a fewhomotopy iterations.

An adaptive viscosity regularization approach for the numerical solution of conservation laws: Application to finite element methods