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 .
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Given a hyperbolic conservation equation with an initial value :
![U_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52116e300c6199496d7b4a60d417ba34a4e569dc)
![{\ displaystyle {\ begin {aligned} \ partial _ {t} U + \ partial _ {x} F (U) & = 0, \\ U (x, 0) & = U_ {0} (x), \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/071ab3f81697a861ebf24b9be78bbc500339d2a7)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d35b4a50773b95b0e1d7e554f080c78f36679af)
![{\ displaystyle F \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c13fd91a344a58a3e65cbdb662518a5979b75c9)
![{\ displaystyle {\ tilde {F}} \ colon \ mathbb {R} ^ {n} \ times \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/211fa4167025267484916b38ca798556af6a9a4e)
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be consistent: for everyone is .![{\ displaystyle V \ in \ mathbb {R} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb29919b6f95ee8727f534e994bb73f91fcc616)
![{\ displaystyle {\ tilde {F}} (V, V) = F (V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15af915c21379d012f8fae31f0f64cc6201b4035)
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be Lipschitz continuous in every argument.
- the numerical approximations have compact support and limited variation : .
![{\ displaystyle TV \ left (U _ {\ Delta t} (\ cdot, t) \ right) <\ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ded7e5788a0de40135d2c6199c1c09ef79ee0e53)
If the numerical approximations now converge:
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so is a weak solution to the initial value problem.
![U](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025)
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