Lax-Wendroff's theorem

The set of Lax-Wendroff states that if the numerical solutions of a hyperbolic conservation equation converge it with a weak solution converge the equation. It is a statement from numerical mathematics that is named after Peter Lax and Burton Wendroff .

sentence

Given a hyperbolic conservation equation with an initial value : ${\displaystyle U_{0}}$

${\displaystyle {\begin{aligned}\partial _{t}U+\partial _{x}F(U)&=0,\\U(x,0)&=U_{0}(x),\end{aligned}}}$

where is the function sought and the exact flow function . The numerical flow function is given. The following must also apply: ${\displaystyle U\colon \mathbb {R} \times \mathbb {R} ^{+}\to \mathbb {R} ^{n}}$ ${\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle {\tilde {F}}\colon \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$

1. ${\displaystyle {\tilde {F}}}$be consistent: for everyone is . ${\displaystyle V\in \mathbb {R} ^{n}}$ ${\displaystyle {\tilde {F}}(V,V)=F(V)}$
2. ${\displaystyle {\tilde {F}}}$be Lipschitz continuous in every argument.
3. the numerical approximations have compact support and limited variation : . ${\displaystyle U_{\Delta t}}$ ${\displaystyle TV\left(U_{\Delta t}(\cdot ,t)\right)<\infty }$

If the numerical approximations now converge:

${\displaystyle \|U_{\Delta t}-U\|_{L^{1}(\mathbb {R} \times \mathbb {R} ^{+})}\longrightarrow 0\quad {\textrm {mit}}\;\Delta t\rightarrow 0}$,

so is a weak solution to the initial value problem. ${\displaystyle U}$

literature

• Randall J. LeVeque : Numerical methods for conservation laws. Birkhäuser, 1992, ISBN 978-3-7643-2723-1 .