Well-Balanced High-Order Finite Volume Methods for Systems of Balance Laws

Abstract

In some previous works, the authors have introduced a strategy to develop well-balanced high-order numerical methods for nonconservative hyperbolic systems in the framework of path-conservative numerical methods. The key ingredient of these methods is a well-balanced reconstruction operator, i.e. an operator that preserves the stationary solutions in some sense. A strategy has been also introduced to modify any standard reconstruction operator like MUSCL, ENO, CWENO, etc. in order to be well-balanced. In this article, the specific case of 1d systems of balance laws is addressed and difficulties are gradually introduced: the methods are presented in the simpler case in which the source term does not involve Dirac masses. Next, systems whose source term involves the derivative of discontinuous functions are considered. In this case, the notion of weak solution is discussed and the Generalized Hydrostatic Reconstruction technique is used for the treatment of singular source terms. A technique to preserve the well-balancedness of the methods in the presence of numerical integration is introduced. The strategy is applied to derive first, second and third order well-balanced methods for Burgers’ equation with a nonlinear source term and for the Euler equations with gravity.

Appendix A

It can be easily checked that, for smooth solutions, (1) with initial condition

$$\begin{aligned} U(x,0) = U_0(x) \end{aligned}$$

is equivalent to the system

$$\begin{aligned} W_t + A(W) W_x = 0, \end{aligned}$$

(81)

where

$$\begin{aligned} W = \left[ \begin{array}{l} U \\ \sigma \end{array} \right] , \quad A(W) = \left[ \begin{array}{cc} J(U) &{} -S(U) \\ 0 &{} 0 \end{array} \right] , \end{aligned}$$

(82)

with initial condition

$$\begin{aligned} W(x,0) = \left[ \begin{array}{c} U_0(x) \\ H(x) \end{array} \right] . \end{aligned}$$

The eigenvalues of A(W) are those of J(U) plus $\lambda ^* = 0$ and an associated eigenvector to $\lambda ^*$ is given by

$$\begin{aligned} R^*(W) = \left[ \begin{array}{c} J(U)^{-1} S(U) \\ 1 \end{array} \right] . \end{aligned}$$

Since the characteristic field $R^*(U)$ is linearly degenerate (as $\lambda ^*$ is constant), if (81) was a system of conservation laws, the Riemann invariants would have to be preserved through the stationary contact discontinuities related to the null eigenvalue: in other words, the limit states would have to belong to the same integral curve of the corresponding linearly degenerate field. These integral curves satisfy:

$$\begin{aligned} \frac{dW}{ds} = R^*(W) \end{aligned}$$

or, equivalently

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{d U}{d s} = J(U)^{-1} S(U) ,\\ \\ \displaystyle \frac{d H}{d s} = 1 , \end{array} \right. \end{aligned}$$

i.e.

$$\begin{aligned} \frac{dU}{dH} = J(U)^{-1} S(U) \end{aligned}$$

what is, with a different notation, equivalent to (36). Therefore, the preservation of the Riemann invariants is equivalent to the admissibility criterion (36)–(37).

Let us remark that, in this case, the preservation of the Riemann invariants through the stationary contact discontinuities, although natural, is not mandatory: other admissibility criteria may lead to different notions of weak solution. For instance, in the case of the Shallow Water system, the jump conditions imposed at the stationary jumps related to the bottom discontinuities in [5] are not equivalent to the preservation of the Riemann invariants.