A $$P_N P_M{-} CPR $$ P N P M - C P R Framework for Hyperbolic Conservation Laws

Abstract

The correction procedure via reconstruction (CPR) method is a discontinuous nodal formulation unifying several well-known methods in a simple finite difference like manner. The $$P_NP_M{-} CPR $$ P N P M - C P R formulation is an extension of $$P_NP_M$$ P N P M or the reconstructed discontinuous Galerkin (RDG) method to the CPR framework. It is a hybrid finite volume and discontinuous Galerkin (DG) method, in which neighboring cells are used to build a higher order polynomial than the solution representation in the cell under consideration. In this paper, we present several $$P_NP_M$$ P N P M schemes under the CPR framework. Many interesting schemes with various orders of accuracy and efficiency are developed. The dispersion and dissipation properties of those methods are investigated through a Fourier analysis, which shows that the $$P_NP_M{-} CPR $$ P N P M - C P R method is dependent on the position of the solution points. Optimal solution points for 1D $$P_NP_M{-} CPR $$ P N P M - C P R schemes which can produce expected order of accuracy are identified. In addition, the $$P_NP_M{-} CPR $$ P N P M - C P R method is extended to solve 2D inviscid flow governed by the Euler equations and several numerical tests are performed to assess its performance.