Motivation

In this work we perform three basic exercises applying high-resolution finite volume methods for hyperbolic problems. We will use one of the so called HRSC (High Resolution Shock Capturing) methods which are suited for the evolution of discontinuities. First we study the advection equation, then the Burgers equation and finally a model for a one dimensional ultra-relativistic fluid. Martin Snajdr provided a software infrastructure to carry out this computations and we implemented a few subroutines and modifications of the basic template.

The theory of numerical methods for nonlinear hyperbolic partial differential equations, or conservation laws, has become one of the great successes of numerical analysis. The development of schemes for nonlinear hyperbolic equations requires an understanding of both numerical analysis and the theory of nonlinear hyperbolic equations. By using knowledge of the structure of the solutions of these equations, methods have been developed that compute highly accurate solutions.

The development of the theory of nonlinear hyperbolic partial differential equations in the last fifty years has been stimulated by the growth in applications such as supersonic aerodynamics, thermonuclear explosions, and oil recovery. In each of these applications the differential equations express the conservation of mass, momentum, and other quantities. A scalar conservation law in one space dimension takes the form,

$$\frac{\partial q}{\partial t}+\frac{\partial F(u(x,t))}{\partial x}=0$$ (1)

where $F(u)$ is a flux function (in general, conservation laws are nonlinear). The increased power of numerical computations has enabled researchers to study ever more complex physical problems governed by conservation laws. A better understanding of the solutions of the differential equations was needed to develop schemes that would compute accurate solutions. Usually the computations of physical phenomena have been beyond the pale of the theory, serving to motivate further work on it.

The theory of conservation laws is a rich part of mathematics. One of the best introductions to this theory is the book by Lax [7]. The basic difficulty with nonlinear-hyperbolic partial differential equations is that the solutions develop singularities, especially discontinuities, which are usually called shock waves or shocks. With the development of these discontinuous solutions there is usually the potential for non-uniqueness of the solutions. The mathematical problem is to introduce restrictions on the class of solutions so that a unique solution exists and so that this solution corresponds to physical phenomena. To restrict the set of solutions, it is required that solutions must be weak solutions, that is, they must be consistent with an integral form of the equations. The integral form is obtained by multiplying the differential equation by a test function and then using integration by parts to remove all differentiation from the solution of the differential equation. However, even among the class of weak solutions, the solution is not uniquely determined.

For particular nonlinear hyperbolic systems arising in applications, there usually are physical principles, such as the second law of thermodynamics [6], that select unique solution from among the weak solutions. In general, the use of similar ``entropy-conditions'' supplies the principle that selects a unique solution from among all weak solutions.

The possibility of non-unique solutions presents difficulties for the development of numerical methods for computing solutions. Indeed, it is easy to construct finite difference schemes for conservation laws that compute incorrect solutions, i.e., solutions not satisfying the correct entropy condition or solutions converging to solutions that are not weak solutions. The basic result on the convergence of solutions of schemes for these equations is the Lax-Wendroff Theorem [8], which states that if a scheme is consistent and conservative, and if the solutions converge as the grid is refined, then the solutions converge to a weak solution of the differential equation. Basically, a conservative scheme is a scheme that satisfies a conservation law analogous to that of the partial differential equation. The development of the theory of conservation laws for multidimensional problems is an area needing more work. Most of the finite difference schemes currently in use for two-dimensional and three-dimensional computations are simple extensions of one-dimensional schemes to multidimensional problems. These schemes have been quite successful in a wide range of computations, but advances in the mathematical theory are required before better schemes can be developed.

This work is just a modest and basic exercise in our training in the field, which has been very valuable and motivating. We encourage the reader to visit our website with most of our results in the form of mpeg animations.