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| The '''Lax–Wendroff method''', named after [[Peter Lax]] and [[Burton Wendroff]], is a [[numerical analysis|numerical]] method for the solution of [[hyperbolic partial differential equation]]s, based on [[finite difference]]s. It is second-order accurate in both space and time. This method is an example of [[temporal discretization|explicit time integration]] where the function that defines governing equation is evaluated at the current time.
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| Suppose one has an equation of the following form:
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| : <math> \frac{\partial f(x,t)}{\partial t}=\frac{\partial g(f(x,t))}{\partial x}\,</math>
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| where ''x'' and ''t'' are independent variables, and the initial state, ƒ(''x'', 0) is given.
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| The first step in the Lax–Wendroff method calculates values for ƒ(''x'', ''t'') at half time steps, ''t''<sub>''n'' + 1/2</sub> and half grid points, ''x''<sub>''i'' + 1/2</sub>. In the second step values at ''t''<sub>''n'' + 1</sub> are calculated using the data for ''t''<sub>''n''</sub> and ''t''<sub>''n'' + 1/2</sub>.
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| First (Lax) step:
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| : <math> \cfrac{f_{i+1/2}^{n+1/2} - \cfrac{f_i^n+f_{i+1}^n}{2}}{(1/2) * \Delta t}=\cfrac{g_{i+1}^n - g_i^n}{\Delta x}.\,</math>
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| Second step:
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| : <math> \cfrac{f_i^{n+1} - f_i^n}{\Delta t}=\cfrac{g_{i+1/2}^{n+1/2} - g_{i-1/2}^{n+1/2}}{\Delta x}.\, </math>
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| This method can be further applied to some systems of partial differential equations.
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| ==References==
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| * {{ cite journal | author = P.D Lax | coauthors = B. Wendroff | year = 1960 | title = Systems of conservation laws | journal = Commun. Pure Appl Math. | volume = 13 | pages = 217–237 | doi = 10.1002/cpa.3160130205 | issue = 2 }}
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| * Michael J. Thompson, ''An Introduction to Astrophysical Fluid Dynamics'', Imperial College Press, London, 2006.
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| *{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 20.1. Flux Conservative Initial Value Problems | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1040 | page=1040}}
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| {{Numerical PDE}}
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| {{DEFAULTSORT:Lax-Wendroff Method}}
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| [[Category:Numerical differential equations]]
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| [[Category:Computational fluid dynamics]]
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| {{mathapplied-stub}}
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