Multidomain Hybrid Direct DG and Central Difference Methods for Viscous Terms in Hyperbolic-Parabolic Equations

A class of multidomain hybrid methods of direct discontinuous Galerkin (DDG) methods and central difference (CD) schemes for the viscous terms is proposed in this paper. Both conservative and nonconservative coupling modes are discussed. To treat the shock wave, the nonconservative coupling mode automatically switch to conservative coupling mode to preserve the conservative property when discontinuities pass through the artificial interface. To maintain the accuracy of the hybrid methods, the Lagrange interpolation polynomials and their derivatives are reconstructed to handle the coupling cells in the DDG subdomain, while the values of ghost points for the CD subdomain are calculated by the approximate polynomials from the DDG methods. The linear stabilities of these methods are demonstrated in detail through von-Neumann analysis. The multidomain hybrid DDG and CD methods are then extended to one- and two-dimensional hyperbolic-parabolic equations. Numerical results validate that the multidomain hybrid methods are high-order accurate in the smooth regions, robust for viscous shock simulations and capable to save computational cost.