Transport-collapse scheme for scalar conservation laws -- initial and boundary value problems
For researchers in hyperbolic conservation laws, this provides a theoretical extension and numerical scheme for boundary value problems, though it is an incremental refinement of prior work.
The paper extends Brenier's transport collapse scheme to heterogeneous scalar conservation laws and initial-boundary value problems, proving convergence to the entropy admissible solution and introducing a new solution concept for boundary problems, supported by numerical examples.
We extend Brenier's transport collapse scheme on the Cauchy problem for heterogeneous scalar conservation laws and initial-boundary value problem for homogeneous scalar conservation laws. It is based on averaging out the solution to the corresponding kinetic equation, and it necessarily converges toward the entropy admissible solution. In the case of initial-boundary value problem, we such a procedure is used to construct a numerical scheme which leads us to a new solution concept for initial-boundary value problem for scalar conservation laws. The concept is a generalization (refinement) of the previous works on initial-boundary value problem. We also provide numerical examples.