Abstract
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 of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.
Highlights
IntroductionLet us consider a PDE system of the form: Ut(x, t) + f (U(x, t))x = S(U(x, t))Hx(x), x ∈ R, t > 0, (1)where U(x, t) takes values on an open convex set Ω ⊂ RN, f : Ω −→ RN is the flux function, S : Ω −→ RN, and H is a continuous known function from R to R (possibly the identity function H(x) = x). It is supposed that system (1) is strictly hyperbolic, that is, D f (U) = ∂f ∂U (U) hasN real distinct eigenvalues r1(U), · · · , rN(U)and associated eigenvectors v1, · · · , vN.Systems of the form (1) have non-trivial stationary solutions that satisfy the ODE system:
Let us consider a PDE system of the form: Ut(x, t) + f (U(x, t))x = S(U(x, t))Hx(x), x ∈ R, t > 0, (1)where U(x, t) takes values on an open convex set Ω ⊂ RN, f : Ω −→ RN is the flux function, S : Ω −→ RN, and H is a continuous known function from R to R (possibly the identity function H(x) = x)
To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects
Summary
Let us consider a PDE system of the form: Ut(x, t) + f (U(x, t))x = S(U(x, t))Hx(x), x ∈ R, t > 0, (1)where U(x, t) takes values on an open convex set Ω ⊂ RN, f : Ω −→ RN is the flux function, S : Ω −→ RN, and H is a continuous known function from R to R (possibly the identity function H(x) = x). It is supposed that system (1) is strictly hyperbolic, that is, D f (U) = ∂f ∂U (U) hasN real distinct eigenvalues r1(U), · · · , rN(U)and associated eigenvectors v1, · · · , vN.Systems of the form (1) have non-trivial stationary solutions that satisfy the ODE system:
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