Lax–Wendroff method: Difference between revisions

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{{Short description|Numerical methods for partial differential equations}}

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.

[[File:Advection_equation_solution_comparison.png | thumb | right | Graphs that show different Lax–Wendroff methods]]

The '''Lax–Wendroff method''', named after [[Peter Lax]] and [[Burton Wendroff]],<ref>{{ cite journal | author1 = P.D Lax | author2 =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 | url = https://apps.dtic.mil/sti/pdfs/ADA385056.pdf | archive-url = https://web.archive.org/web/20170925220837/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA385056 | url-status = live | archive-date = September 25, 2017 }}</ref> 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 the governing equation is evaluated at the current time.

== Definition ==

Suppose one has an equation of the following form:

: <math display="block"> \frac{\partial u(x,t)}{\partial t} + \frac{\partial f(u(x,t))}{\partial x} = 0\,</math>

where ''{{mvar|x''}} and ''{{mvar|t''}} are independent variables, and the initial state, {{math|''u''(''x'',&nbsp; 0)}} is given.

=== Linear case ===

: <math> \frac{\partial u(x,t)}{\partial t}+\frac{\partial f(u(x,t))}{\partial x}=0\,</math>

In the linear case, where {{math|1=''f''(''u'') = ''Au''}}, and {{math|''A''}} is a constant,<ref>{{cite book |last=LeVeque |first=Randall J. |title=Numerical Methods for Conservation Laws |___location=Boston |publisher=Birkhäuser |year=1992 |isbn=0-8176-2723-5 |page=125 |url=http://tevza.org/home/course/modelling-II_2016/books/Leveque%20-%20Numerical%20Methods%20for%20Conservation%20Laws.pdf#page=138 }}</ref>

: <math display="block"> u_i^{n+1} = u_i^n - \frac{\Delta t}{2\Delta x} A\left[ u_{i+1}^{n} - u_{i-1}^{n} \right] + \frac{\Delta t^2}{2\Delta x^2} A^2\left[ u_{i+1}^{n} -2 u_{i}^{n} + u_{i-1}^{n} \right].</math>

where ''x'' and ''t'' are independent variables, and the initial state, u(''x'',&nbsp;0) is given.

Here <math>n</math> refers to the <math>t</math> dimension and <math>i</math> refers to the <math>x</math> dimension.

In the linear case, where '' f(u) = Au '', and ''A'' is a constant,<ref>LeVeque, Randy J. ''Numerical Methods for Conservation Laws", Birkhauser Verlag, 1992, p. 125.</ref>

: <math> u_i^{n+1} = u_i^n - \frac{\Delta t}{2\Delta x} A\left[ u_{i+1}^{n} - u_{i-1}^{n} \right] + \frac{\Delta t^2}{2\Delta x^2} A^2\left[ u_{i+1}^{n} -2 u_{i}^{n} + u_{i-1}^{n} \right].</math>

This linear scheme can be extended to the general non-linear case in different ways. One of them is letting

: <math display="block"> A(u) = f'(u) = \frac{\partial f}{\partial u}</math>

=== Non-linear case ===

: <math> A(u) = f'(u) = \frac{\partial f}{\partial u}</math>

The conservative form of Lax-Wendroff for a general non-linear equation is then:

: <math display="block"> u_i^{n+1} = u_i^n - \frac{\Delta t}{2\Delta x} \left[ f(u_{i+1}^{n})) - f(u_{i-1}^{n}) \right] + \frac{\Delta t^2}{2\Delta x^2} \left[ A_{i+1/2} \left(f(u_{i+1}^{n}) - f(u_{i}^{n})\right) - A_{i-1/2}\left( f(u_{i}^{n})-f(u_{i-1}^{n})\right) \right].</math>

where <math>A_{i\pm 1/2}</math> is the Jacobian matrix evaluated at <math display="inline">\frac{1}{2} (Uu^n_i + Uu^n_{i\pm 1/2})</math>.

== Jacobian free methods ==

The conservative form of Lax-Wendroff for a general non-linear equation is then

To avoid the Jacobian evaluation, use a two-step procedure. What follows is the Richtmyer two-step Lax–Wendroff method.

=== Richtmyer method ===

: <math> u_i^{n+1} = u_i^n - \frac{\Delta t}{2\Delta x} \left[ f(u_{i+1}^{n})) - f(u_{i-1}^{n}) \right] + \frac{\Delta t^2}{2\Delta x^2} \left[ A_{i+1/2}\left(f(u_{i+1}^{n} - f(u_{i}^{n})\right) - A_{i-1/2}\left( f(u_{i}^{n})-f(u_{i-1}^{n})\right) \right].</math>

What follows is the Richtmyer two-step Lax–Wendroff method. The first step in the Richtmyer two-step Lax–Wendroff method calculates values for {{math|''f''(''u''(''x'',&nbsp; ''t''))}} at half time steps, {{math|''t''_{''n''&nbsp; +&nbsp; 1/2}}} and half grid points, {{math|''x''_{''i''&nbsp; +&nbsp; 1/2}}}. In the second step values at {{math|''t''_{''n''&nbsp; +&nbsp; 1}}} are calculated using the data for {{math|''t''_''n''}} and {{math|''t''_{''n''&nbsp; +&nbsp; 1/2}}}.

where <math>A_{i\pm 1/2}</math> is the Jacobian matrix evaluated at <math>\frac{1}{2}(U^n_i + U^n_{i\pm 1/2})</math>.

To avoid the Jacobian evaluation, use a two-step procedure. What follows is the Richtmyer two-step Lax–Wendroff method.

The first step in the Richtmyer two-step Lax–Wendroff method calculates values for f(u(''x'',&nbsp;''t'')) at half time steps, ''t''_{''n''&nbsp;+&nbsp;1/2} and half grid points, ''x''_{''i''&nbsp;+&nbsp;1/2}. In the second step values at ''t''_{''n''&nbsp;+&nbsp;1} are calculated using the data for ''t''_''n'' and ''t''_{''n''&nbsp;+&nbsp;1/2}.

First (Lax) steps:

: <math display="block"> u_{i-+1/2}^{n+1/2} = \frac{1}{2}(u_{i+1}^n + u_{i-1}^n) - \frac{\Delta t}{2\,\Delta x}( f(u_{i+1}^n) - f(u_{i-1}^n) ).,</math>

: <math display="block"> u_{i+-1/2}^{n+1/2} = \frac{1}{2}(u_{i+1}^n + u_{i-1}^n) - \frac{\Delta t}{2\,\Delta x}( f(u_{i+1}^n) - f(u_{i-1}^n) ),.</math>

: <math> u_{i-1/2}^{n+1/2}= \frac{1}{2}(u_{i}^n + u_{i-1}^n) - \frac{\Delta t}{2\,\Delta x}( f(u_{i}^n) - f(u_{i-1}^n) ).</math>

Second step:

: <math display="block"> u_i^{n+1} = u_i^n - \frac{\Delta t}{\Delta x} \left[ f(u_{i+1/2}^{n+1/2}) - f(u_{i-1/2}^{n+1/2}) \right].</math>

=== MacCormack method ===

: <math> u_i^{n+1} = u_i^n - \frac{\Delta t}{\Delta x} \left[ f(u_{i+1/2}^{n+1/2}) - f(u_{i-1/2}^{n+1/2}) \right].</math>

{{Main|MacCormack method}}

Another method of this same type was proposed by MacCormack. MacCormack's method uses first forward differencing and then backward differencing:

First step:

: <math display="block"> u_{i}^{*}= u_{i}^n - \frac{\Delta t}{\Delta x}( f(u_{i+1}^n) - f(u_{i}^n) ).</math>

Second step:

: <math display="block"> u_i^{n+1} = \frac{1}{2} (u_{i}^n + u_{i}^*) - \frac{\Delta t}{2 \Delta x} \left[ f(u_{i}^{*}) - f(u_{i-1}^{*}) \right].</math>

Alternatively,

First step:

: <math display="block"> u_{i}^{*} = u_{i}^n - \frac{\Delta t}{\Delta x}( f(u_{i}^n) - f(u_{i-1}^n) ).</math>

Second step:

: <math display="block"> u_i^{n+1} = \frac{1}{2} (u_{i}^n + u_{i}^*) - \frac{\Delta t}{2 \Delta x} \left[ f(u_{i+1}^{*}) - f(u_{i}^{*}) \right].</math>

==References==

{{Reflist}}

* {{ cite journal | author = P.D Lax |author2=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 }}

* Michael J. Thompson, ''An Introduction to Astrophysical Fluid Dynamics'', Imperial College Press, London, 2006.

*{{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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[[Category:Numerical differential equations]]

[[Category:Computational fluid dynamics]]

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