Unsplit numerical schemes for hyperbolic systems of conservation laws with source terms

In this thesis, a new method for the design of unsplit numerical schemes for hyperbolic systems of conservation laws with source terms is developed. Appropriate curves in space-time are introduced, along which the conservation equations decouple to the characteristic equations of the corresponding one-dimensional homogeneous system. The local geometry of these curves depends on the source terms and the spatial derivatives of the solution vector. Numerical integration of the characteristic equations is performed on these curves. In the first chapter, a scalar conservation law with a stiff, nonlinear source term is studied using the proposed unsplit scheme. Various tests are made, and the results are compared with the ones obtained by conventional schemes. The effect of the stiffness of the source term is also examined. In the second chapter, the scheme is extended to the one- dimensional, unsteady Euler equations for compressible, chemically-reacting flows. A numerical study of unstable detonations is performed. Detonations in the regime of low overdrive factors are also studied. The numerical simulations verify that the dynamics of the flow-field exhibit chaotic behavior in this regime. The third chapter deals with the development and implementation of the unsplit scheme, far the two- dimensional, reactive Euler equations. In systems with more than two independent variables there are one- parameter families of curves, forming manifolds in space- time, along which the one-dimensional characteristic equations hold. The local geometry of these manifolds and their position relative to the classical characteristic rays are studied. These manifolds might be space-like or time-like, depending an the local flow gradients and the source terms. In the fourth chapter a numerical study of two- dimensional detonations in performed. These flows are intrinsically unstable and produce very complicated patterns, such as cellular structures and vortex sheets. The proposed scheme appears to be capable of capturing many of the the important details of the flow-fields. Unlike traditional schemes, no explicit artificial- viscosity mechanisms need to be used with the proposed scheme