Coupling Two Scalar Conservation Laws via Dafermos' Self-Similar Regularization


We are interested in the problem of coupling two scalar conservation laws with distinct flux-functions. This problem arises, for instance, in modeling fluid flows in media with discontinuous porosity and has important possible applications in the numerical computation of a singular pressure drop. This problem is also well-known to exhibit several technical difficulties due to the presence of nonconservative terms and to the resonant behavior of the system of equations. We present here a global approach consisting of two scalar problems in a half-space coupled through an algebraic jump relation. We view this problem as a $2 times 2$ system of conservation laws, and introduce a viscous regularizationÌ la Dafermos. We establish that this approximation converges as the viscosity tends to zero and we analyze the structure of the entropy solutions constructed in this way.

Numerical Mathematics and Advanced Applications