Upwind differencing schemes for hyperbolic conservation laws with source terms.

*(English)*Zbl 0626.65086
Nonlinear hyperbolic problems, Proc. Adv. Res. Workshop, St. Etienne/France 1986, Lect. Notes Math. 1270, 41-51 (1987).

[For the entire collection see Zbl 0623.00008.]

The present work contributes to the study of non-homogeneous problems in one dimension. We particularly have in mind problems governed by the Euler equations with non-vanishing source terms on the right hand side. These could describe the effects on the flow of area variation, chemical reaction, mass or energy release, or interaction with other material, as in a dusty gas. In all cases the equations to be solved are (1) \(u_ t+F_ x=\underset \tilde{} Q\) where u, F are the vectors representing respectively the conserved quantities and the fluxes, and \(Q=(u,x)\) is the source term. Some numerical aspects of equation (1) are discussed in the simplest case, that of a linear scalar equation. The discussion is extended to systems, and to non-linear systems. High-order schemes are derived, and an example is given.

The present work contributes to the study of non-homogeneous problems in one dimension. We particularly have in mind problems governed by the Euler equations with non-vanishing source terms on the right hand side. These could describe the effects on the flow of area variation, chemical reaction, mass or energy release, or interaction with other material, as in a dusty gas. In all cases the equations to be solved are (1) \(u_ t+F_ x=\underset \tilde{} Q\) where u, F are the vectors representing respectively the conserved quantities and the fluxes, and \(Q=(u,x)\) is the source term. Some numerical aspects of equation (1) are discussed in the simplest case, that of a linear scalar equation. The discussion is extended to systems, and to non-linear systems. High-order schemes are derived, and an example is given.

##### MSC:

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |