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

Abstract

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.

Construction of Second-Order TVD Schemes for Nonhomogeneous Hyperbolic Conservation Laws

“…This new contribution has the same form as the propagation speeds introduced by Papalexandris et al in [21].…”

“…Motivated by the fact that if there is a source term the Riemann invariants are not constant along the characteristic trajectories, Papalexandris et al [21] have described the curves in space-time along which the characteristic system holds for the nonhomogeneous case. This new decomposition is used by the authors in the design of efficient unsplit algorithms for the numerical integration of the systems of hyperbolic conservation laws with source terms.…”

Second-order Godunov-type scheme for reactive flow calculations on moving meshes

“…The dimensionless parameters by reference to the uniform state ahead the detonation shock, moving to the right, are γ=1.2, q o =50, E + =50, the degree of overdrive f = (D/D CJ ) 2 =1.6, where D is the shock speed, K o =230.75, R=1, (u, p, ρ, Z) unb =(0, 1, 1, 1). This problem was also simulated numerically in [25,8,44,42,29].…”

“…In [20], the detonation process is simulated on the Lagrangian mesh. Space-time paths are introduced in [42] on which the equations are reduced to the canonical form about the "new" Riemann invariants. All the above methods are of Godunov-type (except [16], where the random choice method is used), i.e.…”

Front Tracking for Scalar Balance Equations

“…Roe [65] suggested to apply high-resolution upwind schemes to a modified flux that includes the source term ("upwinding of the source term"). In the same spirit, Bermúdez and Vazquez [4] (see also [3,71,37,16,9,59,60] for related papers) extended some upwind (flux-difference splitting and flux-vector splitting) schemes to hyperbolic systems of conservation laws with source terms (shallow water equations). To ensure that the numerical schemes approximate to high order steady-state solutions, they introduced the so-called "conservation property" (this property can be seen related to Liu's "piecewise stationary discretization" as well as LeRoux and Greenberg's notion of well-balanced schemes [23]).…”