Partial differential equations Definition

Glimm scheme

Builds on Conservation law Riemann problem Hyperbolic conservation law

Definition

Let $u_t + F(u)_x = 0$ be a strictly hyperbolic system of conservation laws, with $F \colon \Omega \to \R^m$ smooth on an open set $\Omega \subseteq \R^m$ . Assume that every Riemann problem with states in $\Omega$ close enough to each other has an entropy solution. Let $\Lambda$ be an upper bound for the absolute values of the eigenvalues of $DF(u)$ on $\Omega$ . Fix a space step $\Delta x > 0$ and a time step $\Delta t > 0$ with $\Lambda \Delta t \le \Delta x / 2$ . Set $x_j = j \Delta x$ for $j \in \Z$ and $t_n = n \Delta t$ for $n \ge 0$ . Fix a sequence $(\theta_n)_{n \ge 1}$ in $(0, 1)$ .

The Glimm scheme defines an approximate solution $u^\Delta$ of the initial value problem with data $u_0$ as follows.

At $t = 0$ , set $u^\Delta(x, 0) = u_0(x_j)$ for $x_{j-1} < x \le x_j$ .
If $u^\Delta(\cdot, t_n)$ is constant on each cell $(x_{j-1}, x_j]$ , then on $t_n \le t < t_{n+1}$ let $u^\Delta$ be the exact solution of the Riemann problems at the points $x_j$ .
At $t = t_{n+1}$ , set $u^\Delta(x, t_{n+1}) = u^\Delta(x_{j-1} + \theta_{n+1} \Delta x,\, t_{n+1} - 0)$ for $x_{j-1} < x \le x_j$ .

When the $\theta_n$ are chosen independently and uniformly at random, it is also called the random choice method.

Remarks

The bound $\Lambda \Delta t \le \Delta x/2$ is a Courant–Friedrichs–Lewy condition. It ensures that waves from neighbouring points $x_j$ do not meet before time $t_{n+1}$ . So in step 2 the solution on the strip is exactly the union of the local Riemann solutions.

Glimm proved that, if the total variation of $u_0$ is small enough, the total variation of $u^\Delta(\cdot, t)$ stays bounded uniformly in $\Delta x$ and $t$ . For almost every random sequence $(\theta_n)$ , as $\Delta x \to 0$ with $\Delta t / \Delta x$ fixed, a subsequence of the approximations converges to a weak solution. T.-P. Liu later showed that any equidistributed sequence $(\theta_n)$ gives convergence.

Example: for the linear equation $u_t + a u_x = 0$ with $0 < a$ and $a \Delta t \le \Delta x/2$ , the Riemann solution at $x_j$ moves the jump to $x_j + a(t - t_n)$ . At $t_{n+1}$ , a cell takes its old value if $\theta_{n+1} \Delta x > a \Delta t$ , and the value of its left neighbour otherwise.

Non-example: step 3 with the cell average instead of a sampled value is not the Glimm scheme. It is Godunov's scheme, which smears shocks over several cells, while sampling keeps them sharp.

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Tag	`5JQG4S5JVH`
URI	`m2://5JQG4S5JVH`
Kind	definition
Writing	W0 stub
MSC	35L65, 65M06, 35L45
Defines	Glimm scheme
Uses	Conservation law, Riemann problem, Hyperbolic conservation law

Glimm scheme

Definition

Remarks

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