The Construction of Discretely Conservative Finite Volume Schemes that Also Globally Conserve Energy or Entropy

Abstract

This work revisits an idea that dates back to the early days of scientific computing, the energy method for stability analysis. It is shown that if the scalar non-linear conservation law

$$\frac{\partial u}{\partial t}+\frac{\partial}{\partial x}f(u)=0$$

is approximated by the semi-discrete conservative scheme

$$\frac{du_{j}}{dt}+\frac{1}{\Delta x}\left(f_{j+\frac{1}{2}}-f_{j-\frac{1}{2}}\right)=0$$

then the energy of the discrete solution evolves at exactly the same rate as the energy of the true solution, provided that the numerical flux is evaluated by the formula

$$f_{j+\frac{1}{2}}=\int_{0}^{1}f(\hat{u})d\theta,$$

where

$$\hat{u}(\theta)=u_{j}+\theta(u_{j+1}-u_{j}).$$

With careful treatment of the boundary conditions, this provides a path to the construction of non-dissipative stable discretizations of the governing equations. If shock waves appear in the solution, the discretization must be augmented by appropriate shock operators to account for the dissipation of energy by the shock waves. These results are extended to systems of conservation laws, including the equations of incompressible flow, and gas dynamics. In the case of viscous flow, it is also shown that shock waves can be fully resolved by non-dissipative discretizations of this type with a fine enough mesh, such that the cell Reynolds number ≤2.