Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis these is well-balanced reconstruction operator. Moreover, they procedure modify any standard operator, like MUSCL, ENO, CWENO, etc., in order be well-balanced. This strategy involves non-linear problem at every cell time step consists finding solution whose average given value. recent paper, fully method presented where problems solved are interpreted as control problems. goal this paper introduce new solve local based on application collocation RK methods. Special care put analyze effects computing averages and source terms using quadrature formulas. A general which allows us deal with resonant also introduced. To check efficiency well-balance property, have been applied number tests, ranging from easy academic systems consisting Burgers equation shallow water equations—without Manning friction—or Euler equations gas dynamics gravity effects.