A strategy to implement Dirichlet boundary conditions in the context of ADER finite volume schemes. One-dimensional conservation laws

TL;DR

This paper presents a novel boundary treatment strategy for ADER finite volume schemes solving conservation laws, ensuring high-order accuracy at boundaries by using reverse hyperbolic auxiliary problems without Taylor series expansions.

Contribution

The paper introduces a new method for implementing Dirichlet boundary conditions in ADER schemes using reverse problems, avoiding Taylor series and Lax-Wendroff procedures, and achieving up to fifth-order accuracy.

Findings

Method preserves scheme accuracy at boundaries.

Achieves up to fifth-order accuracy in space and time.

Numerical results confirm effectiveness of the approach.

Abstract

ADER schemes are numerical methods, which can reach an arbitrary order of accuracy in both space and time. They are based on a reconstruction procedure and the solution of generalized Riemann problems. However, for general boundary conditions, in particular of Dirichlet type, a lack of accuracy might occur if a suitable treatment of boundaries conditions is not properly carried out. In this work the treatment of Dirichlet boundary conditions for conservation laws in the context of ADER schemes, is concerned. The solution of generalized Riemann problems at the extremes of the computational domain, provides the correct influence of boundaries. The reconstruction procedure, for data near to the boundaries, demands for information outside the computational domain, which is carried out in terms of ghost cells, which are provided by using the numerical solution of auxiliary problems. These auxiliary problems are hyperbolic and they are constructed from the conservation laws and the information at boundaries, which may be partially or totally known in terms of prescribed functions. The evolution of these problems, unlike to the usual manner, is done in space rather than in time due to that these problems are named here, {\it reverse problems}. The methodology can be considered as a numerical counterpart of the inverse Lax-Wendroff procedure for filling ghost cells. However, the use of Taylor series expansions, as well as, Lax-Wendroff procedure, are avoided. For the scalar case is shown that the present procedure preserve the accuracy of the scheme which is reinforced with some numerical results. Expected orders of accuracy for solving conservation laws by using the proposed strategy at boundaries, are obtained up to fifth-order in both space and time.