Numerical analysis Time-dependent partial differential equations Proposition

Roe solver

Builds on Riemann solver Hyperbolic conservation law Riemann problem Diagonalisable matrix Eigenvalue Shock wave

Statement

Let $\Omega \subseteq \R^m$ be open and let $F \colon \Omega \to \R^m$ be the $C^1$ flux of a hyperbolic conservation law $u_t + F(u)_x = 0$ . A Roe matrix is a map $(u_l, u_r) \mapsto A(u_l, u_r)$ from $\Omega \times \Omega$ to the real $m \times m$ matrices with three properties:

$F(u_r) - F(u_l) = A(u_l, u_r)(u_r - u_l)$ for all $u_l, u_r \in \Omega$ ;
$A(u_l, u_r)$ is diagonalisable with real eigenvalues;
$A(u, u) = F'(u)$ for all $u \in \Omega$ .

Write $A = A(u_l, u_r) = R \Lambda R^{-1}$ with $\Lambda$ diagonal, and $\abs{A} = R \abs{\Lambda} R^{-1}$ . The Roe flux is $H(u_l, u_r) = \tfrac{1}{2} \big( F(u_l) + F(u_r) \big) - \tfrac{1}{2} \abs{A} (u_r - u_l).$ Then:

$H(u_l, u_r) = F(u_l) + A (w_0 - u_l) = F(u_r) - A (u_r - w_0)$ , and $H(u, u) = F(u)$ . Here $w_0$ is the value on the line $x = 0$ of the solution of the linear Riemann problem $w_t + A w_x = 0$ with data $u_l, u_r$ . So $H$ is a consistent Riemann solver.
Let $u_r \ne u_l$ with $F(u_r) - F(u_l) = s (u_r - u_l)$ for a number $s$ . Then $u_r - u_l$ is an eigenvector of $A$ for the eigenvalue $s$ . The linear Riemann problem is then solved by the single jump from $u_l$ to $u_r$ with speed $s$ .

Remarks

Write $u_r - u_l = \sum_k \alpha_k r_k$ in eigenvectors $r_k$ of $A$ with eigenvalues $\lambda_k$ . The linear Riemann problem moves each component $\alpha_k r_k$ with speed $\lambda_k$ , so $w_0 = u_l + \sum_{\lambda_k < 0} \alpha_k r_k$ . Property 1 of the matrix gives $\tfrac12 (F(u_l) + F(u_r)) = F(u_l) + \tfrac12 \sum_k \lambda_k \alpha_k r_k$ , and subtracting $\tfrac12 \sum_k \abs{\lambda_k} \alpha_k r_k$ leaves $F(u_l) + \sum_{\lambda_k < 0} \lambda_k \alpha_k r_k = F(u_l) + A(w_0 - u_l)$ . Claim 2 is property 1 of the matrix applied to the jump condition.

In the scalar case $m = 1$ the Roe matrix is the number $A = \dfrac{f(u_r) - f(u_l)}{u_r - u_l}$ for $u_r \ne u_l$ , and $A(u, u) = f'(u)$ . For the Euler equations of gas dynamics, Roe found a Roe matrix in closed form. It is the flux Jacobian at averages weighted by $\sqrt{\rho_l}$ and $\sqrt{\rho_r}$ , such as $\tilde v = \dfrac{\sqrt{\rho_l} v_l + \sqrt{\rho_r} v_r}{\sqrt{\rho_l} + \sqrt{\rho_r}}$ for the velocity.

The Roe solver can keep entropy-violating jumps at transonic rarefactions, where an eigenvalue changes sign. For $f(u) = u^2/2$ with $u_l = -1$ and $u_r = 1$ one gets $A = 0$ and $H = \tfrac12$ , so the scheme keeps the initial jump although the entropy solution is a rarefaction fan. An entropy fix, such as the one of Harten, changes $\abs{\lambda_k}$ near zero to remove this defect.

Going deeper

For a student

The Roe flux is a building block of Godunov-type schemes and of MUSCL schemes, where it is evaluated at the reconstructed interface states. Its upwind part $-\tfrac12 \abs{A}(u_r - u_l)$ acts as numerical viscosity: each wave family is damped in proportion to the size of its speed. This keeps stationary shocks and contact discontinuities sharp, but makes the fix near sonic points necessary. Other approximate solvers, such as HLL, trade sharpness for robustness and need only estimates of the wave speeds.

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Tag	`TNM5CFSZM6`
URI	`m2://TNM5CFSZM6`
Kind	statement, proposition
Writing	W0 stub
MSC	65M08, 35L65
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About	Riemann solver
Uses	Riemann solver, Hyperbolic conservation law, Riemann problem, Diagonalisable matrix, Eigenvalue, Shock wave