A cell centered polynomial basis for efficient Galerkin predictors in the context of ADER finite volume schemes. The one dimensional case

Abstract

In this paper, a family of high-order space-time polynomials in the context of continuous and discontinuous Galerkin methods is proposed. The resulting Galerkin schemes are used as a building block in ADER methods for solving one-dimensional hyperbolic balance laws, which can handle stiff source terms. The space-time polynomial basis is constructed as the tensor product of spatial and temporal polynomials. Temporal polynomials are constructed as conventional Lagrange polynomials on a set of temporal nodes, which is formed by Gaussian quadrature points of suitable order. To build the spatial polynomials we propose a set of spatial nodes, the number of these nodes are about a half of those required by conventional Lagrange polynomials.

Then, polynomials and their first derivatives are imposed to be nodal on the set of spatial nodes, it generates two family of degrees of freedom associated with polynomials and their derivatives. The procedure generates even numbers of polynomials. The degrees of freedom of the space-time polynomial solution, resulting from Galerkin approaches are obtained from a system of algebraic equations, which are coupled only in the flux and gradients of fluxes. It allows us to construct an efficient nested-type iteration procedure involving only the source terms and the gradients of source terms, where the set of degrees of freedom are decoupled. Only a few number of iterations are required to get the expected accuracy. Several test cases are solved to evidence the ability of the present scheme for solving hyperbolic balance laws. Expected theoretical orders of accuracy are obtained up to the fourth order in both space and time, using generous CFL numbers.