Lax-Wendroff | zeracuse

The shock wave propagation equation and numerical approach. The governing equation is:

$\frac{\partial \mathbf{q}}{\partial x}+\frac{\partial \mathbf{F}}{\partial x} = 0$

with: $\mathbf{q}=\left[ \rho,\rho u, p/\gamma(\gamma-1) + \frac{\rho}{2}u^2 \right]^T$ and $\mathbf{F}=\left[ \rho u,\rho u^2 +p/\gamma, [p/(\gamma-1)+ 0.5\rho u^2]u \right]^T$

where $\rho,p,u$ are the (non-dimensional) density, pressure and velocity.

FINITE VOLUME METHODS

The functions of interest (the ones expanded with Taylor formula) give a big remainder that is not smaller than the first terms. Moreover it is necessary to conserve the integral of such functions along the computational domain. These demands lead to a design of FV methods that are basically arise from Finite Difference methods. The physical domain is considered to be divided into cells. Between time $t^n$ and time $t^{n+1}$ , the increment of a physical (extensive) quantity, let us say the mass in cell $C_{j}$ , denoted by:

$mass_{j} = meas(C_{j}) density_{j}$

is given by the sum of flux $latex flux_{jk}$ exchanged with each neighbouring cell $C_{j}$ :

$mass_{j} = mass_{j} + \sum_{k\in N(j)} flux_{jk}$ (1)

Conservation of total mass is ensured by the equality:

$flux_{jk} = - flux_{kj}$ (2)

In the case of linear conservation laws, this may be written, for example, as:

$meas(C_{j}) u^{n+1} = meas(C_{j}) u^{n+1} + \sum_{k\in N(j)}(t^{n+1} - t^{n}) Hv(u_{j}, u_{k})$

in which the elementary flux is assumed to satisfy a consistency equality and to approximate the normal flux of $uV$ through the common interface of the two cells:

$Hv(u_{j}, u{j}) = \int_{\partial C_{j}\cap \partial C_{j}} Vnu_{j} d \sigma$

The Lax-Wendroff theorem for steady linear case can be summarized as:

If the scheme is conservtive according to the flux definition (1),(2), and if the discrete solution converges to a limit, then the limit is a weak solution ( i.e a distribution satisfying the equations).

Accuaracy is influenced by the specific method of discretization and cell type. In cell-center schemes primal cells give the partition. In vertex-centered schemes a dual partition is used for flux-derivation.

Lax-Wendroff scheme was intruduced as the first stable three-point scheme applicable for compressible flow problems. It is commonly used at hyperbolic partial differential equations. It is second-order accurate in both space and time. This method is an example of explicit time integration where the function that defines governing equation is evaluated at the current time. The first step in the Lax–Wendroff method calculates values for the quantity at half time steps, $t_{n+1/2}$ and half grid points, $x_{j+1/2}$ . In the second step values at $t_{n+1}$ are calculated using the data for $t_{n}$ and $t_{n+1/2}$ .