Gino I. Montecinos

Dinshaw S. Balsara, Jiequan Li, Gino I. Montecinos

The Riemann problem, and the associated generalized Riemann problem, are increasingly seen as the important building blocks for modern higher order Godunov-type schemes. In the past, building a generalized Riemann problem solver was seen as an intricately mathematical task for complicated physical or engineering problems because the associated Riemann problem is different for each hyperbolic system of interest. This paper changes that situation. The HLLI Riemann solver is a recently-proposed Riemann solver that is universal in that it is applicable to any hyperbolic system, whether in conservation form or with non-conservative products. The HLLI Riemann solver is also complete in the sense that if it is given a complete set of eigenvectors, it represents all waves with minimal dissipation. It is, therefore, very attractive to build a generalized Riemann problem solver version of the HLLI Riemann solver. This is the task that is accomplished in the present paper. We show that at second order, the generalized Riemann problem version of the HLLI Riemann solver is easy to design. Our GRP solver is also complete and universal because it inherits those good properties from original HLLI Riemann solver. We also show how our GRP solver can be adapted to the solution of hyperbolic systems with stiff source terms. Our generalized HLLI Riemann solver is easy to implement and performs robustly and well over a range of test problems. All implementation-related details are presented. Results from several stringent test problems are shown. These test problems are drawn from many different hyperbolic systems, and include hyperbolic systems in conservation form; with non-conservative products; and with stiff source terms. The present generalized Riemann problem solver performs well on all of them.

Gino I. Montecinos

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.

Gino I. Montecinos

Analytic solutions for Burgers equations with source terms, possibly stiff, represent an important element to assess numerical schemes. Here we present a procedure, based on the characteristic technique to obtain analytic solutions for these equations with smooth initial conditions.