# 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:

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

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