Unsplit Schemes for Hyperbolic Conservation Laws with Source Terms in One Space Dimension

Abstract

The present work is concerned with the extension of the theory of characteristics to conservation laws with source terms in one space dimension, such as the Euler equations for reacting flows. New spacetime curves are introduced on which the equations decouple to the characteristic set of O.D.E's for the corresponding homogeneous laws, thus allowing the introduction of functions analogous to the Riemann Invariants. The geometry of these curves depends on the spatial gradients for the solution. This particular decomposition can be used in the design of efficient unsplit algorithms for the numerical integration of the equations. As a first step, these ideas are implemented for the case of a scalar conservation law with a nonlinear source term. The resulting algorithm belongs to the class of MUSCL-type, shock-capturing schemes. Its accuracy and robustness are checked through a series of tests. The aspect of the stiffness of the source term is also studied. Then, the algorithm is generalized for a system of hyperbolic equations, namely the Euler equations for reacting flows. An extensive numerical study of unstable detonations is performed.