On the theoretical foundation of overset grid methods for hyperbolic problems: Well-posedness and conservation

On the Theoretical Foundation of Overset Grid Methods for Hyperbolic Problems II: Entropy Bounded Formulations for Nonlinear Conservation Laws

“…To show that entropy is preserved, we show (41) with the equality, using the Tadmor jump condition, (A.11). For each component, The penalty terms are therefore proportional to the difference between an entropy conserving flux for the specific system of equations and the conservative flux, rather than being directly proportional to the difference in the states, as was the case for linear problems considered in Paper I [14]. In the case of the Burgers equation, however, we show in the following example that the penalty terms can be written that way, but with a solution-dependent coefficient that depends on the two states.…”

“…provided that the penalty matrices are chosen to be positive definite. The relationship between the penalty matrices also ensures conservation [14].…”

“…In a recent paper, [14], hereinafter called Paper I, we studied the overset domain problems for linear hyperbolic systems. We showed that in general, when one cannot characteristically decouple the solutions in the overlap regions, the coupling between the domains must be two-way.…”

On the theoretical foundation of overset grid methods for hyperbolic problems II: Entropy bounded formulations for nonlinear conservation laws

“…where α is either b or c. Note that the penalty is not applied to the overlap region, see Paper I [14] for a discussion.…”

“…In a recent paper, [14], hereinafter called Paper I, we studied the overset domain problems for linear hyperbolic systems. We showed that in general, when one cannot characteristically decouple the solutions in the overlap regions, the coupling between the domains must be two-way for the problem to be well-posed.…”