Stable and convergent residual distribution for time-dependent conservation laws

Abstract

We consider the discretization of the time dependent hyperbolic problem

$$\frac{\partial{u}}{\partial{t}} + \nabla \cdot \mathbf{\mathcal{F}}(\mathbf{u}) = {0} \mathrm{on} \mathit\Omega \times [0,t_f] \subset \mathbb{R}^2 \times \mathbb{R}^+$$

((1))

on unstructured grids. We present residual distribution $(\mathcal{RD})$ schemes which (i) give non-oscillatory solutions, (ii) are second order accurate by construction, and (iii) lead to well-posed algebraic problems, that is, they ultimately lead to linear systems Ax = y, with A invertible. How to construct nonlinear $(\mathcal{RD})$ satisfying (i) and (ii) is known for some time [3]. However, it is the satisfaction of (iii) that ensures that a (unique) discrete solution exists, and that second order of accuracy is actually obtained in practice (convergence).