The generalised Riemann problem: the basis of ADER schemes

Advection-reaction type partial differential equations model a wide variety of phenomena in several disciplines in physics, chemistry, environmental sciences, geometry, financial mathematics and many others. Generally, these non-linear inhomo-geneous equations must be solved in complicated multi-dimensional domains and thus analytical solutions are only available under very special circumstances. Situations in which exact solutions are available include Riemann problems. Conventionally, the Riemann problem for a system of conservation laws in two independent variables Ü and Ø is the initial value problem for the system with initial conditions consisting of two constant states separated by a discontinuity at the origin Ü ¼ ; for background see [3] and [8], for example. Ben-Artzi and Falcovitz [1] and others have generalised the concept of Riemann problem by admitting initial conditions that are linear functions in Ü, separated by a discontinuity at Ü ¼. We introduce the notation ÊÈ ½ to denote this generalisation of the conventional Riemann problem, denoted by ÊÈ ¼. Here we extend further the concept of generalised Riemann problem in two respects, the first concerns initial conditions and the second concerns the type of governing equations. As to initial conditions, we admit-th order polynomial functions of Ü and denote the corresponding generalised Riemann problem by ÊÈ. The most general case is that in which the initial conditions are two arbitrary but infinitely differentiable functions of Ü, with the corresponding generalised Riemann problem denoted by ÊÈ ½. Concerning the governing differential equations, in addition to pure non-linear advection, we include here reaction-like terms; these source terms are assumed to be arbitrary but sufficiently smooth algebraic functions of the unknowns. We propose a semi-analytical method of solution of the generalised Riemann problem ÊÈ ½ for non-linear advection-reaction partial differential equations. The method gives the solution at Ü ¼ at a time , assumed to be sufficiently small, in terms of a time Taylor series expansion at Ü ¼ about Ø ¼. The leading term in this expansion is the exact solution of a conventional non-linear Riemann problem, ÊÈ ¼ , with piece-wise constant initial conditions. All remaining terms in the expansion have coefficients that are time derivatives of the solution; these time derivatives are replaced by spatial derivatives by repeated use of the differential equations, a technique known as the Lax-Wendroff procedure [4]. It is then shown that all spatial derivatives in the expansion obey inhomogeneous advection equations. As derivative values are required …