A Discontinuous Petrov-Galerkin Method for One-Dimensional Hyperbolic Conservation Law Equations Based on HWENO Limiter

Abstract

The discontinuous Petrov-Galerkin method (DPG) is a new type of numerical solution to the hyperbolic conservation law equations, which distinguishes itself from the DG method by its high accuracy based on compact stencils. However, in order to overcome the unphysical oscillations of the higher order linear schemes near the large gradient solution, the DPG method often needs to incorporate limiter functions to obtain a high-resolution image of the numerical solution. This paper attempts to combine HWENO as limiter function with DPG to solve the discontinuous initial value problems for the hyperbolic conservation law equations. The single-step high-accuracy SSP Runge-Kutta method is used for time discretization, and a new HWENO-based process is used as the limiter of the RKDPG methods, which requires only one reconstruction to complete the update of the higher-order moments without calculating the linear weight coefficients. . Since the accuracy does not meet the design requirements, the HWENO limiter above is partially improved for the identification of the problem cells in the HWENO limiter above, with the original numerical solution used at the smooth solution. This paper only gives the calculation results of P1 ~P3, and the limiter is also applicable to the DPG method for higher elements. Numerical examples show that the HWENO limiter can effectively suppress non-physical oscillations in the problem cells and keep the original accuracy at the non-problem cells. The high accuracy and compactness characteristics of the DPG method are maintained. The numerical calculation solutions are efficient and accurate.