Lax–Wendroff method#Richtmyer method

Suppose one has an equation of the following form:

$$\backslash frac\{\backslash partial\; u(x,t)\}\{\backslash partial\; t\}\; +\; \backslash frac\{\backslash partial\; f(u(x,t))\}\{\backslash partial\; x\}\; =\; 0$$

where {{mvar|x}} and {{mvar|t}} are independent variables, and the initial state, {{math|u(x, 0)}} is given.

In the linear case, where {{math|1=f(u) = Au}}, and {{math|A}} is a constant,{{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 }}

$$u\_i^\{n+1\}\; =\; u\_i^n\; -\; \backslash frac\{\backslash Delta\; t\}\{2\backslash Delta\; x\}\; A\backslash left[\; u\_\{i+1\}^\{n\}\; -\; u\_\{i-1\}^\{n\}\; \backslash right]\; +\; \backslash frac\{\backslash Delta\; t^2\}\{2\backslash Delta\; x^2\}\; A^2\backslash left[\; u\_\{i+1\}^\{n\}\; -2\; u\_\{i\}^\{n\}\; +\; u\_\{i-1\}^\{n\}\; \backslash right].$$

Here $n$ refers to the $t$ dimension and $i$ refers to the $x$ dimension.

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

$$A(u)\; =\; f\text{'}(u)\; =\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; u\}$$

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

$$u\_i^\{n+1\}\; =\; u\_i^n\; -\; \backslash frac\{\backslash Delta\; t\}\{2\backslash Delta\; x\}\; \backslash left[\; f(u\_\{i+1\}^\{n\})\; -\; f(u\_\{i-1\}^\{n\})\; \backslash right]\; +\; \backslash frac\{\backslash Delta\; t^2\}\{2\backslash Delta\; x^2\}\; \backslash left[\; A\_\{i+1/2\}\; \backslash left(f(u\_\{i+1\}^\{n\})\; -\; f(u\_\{i\}^\{n\})\backslash right)\; -\; A\_\{i-1/2\}\backslash left(\; f(u\_\{i\}^\{n\})-f(u\_\{i-1\}^\{n\})\backslash right)\; \backslash right].$$

where $A\_\{i\backslash pm\; 1/2\}$ is the Jacobian matrix evaluated at $\backslash frac\{1\}\{2\}\; (u^n\_i\; +\; u^n\_\{i\backslash pm\; 1\})$.

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 {{math|f(u(x, t))}} at half time steps, {{math|t_{n + 1/2}}} and half grid points, {{math|x_{i + 1/2}}}. In the second step values at {{math|t_{n + 1}}} are calculated using the data for {{math|t_n}} and {{math|t_{n + 1/2}}}.

First (Lax) steps:

$$u\_\{i+1/2\}^\{n+1/2\}\; =\; \backslash frac\{1\}\{2\}(u\_\{i+1\}^n\; +\; u\_\{i\}^n)\; -\; \backslash frac\{\backslash Delta\; t\}\{2\backslash ,\backslash Delta\; x\}(\; f(u\_\{i+1\}^n)\; -\; f(u\_\{i\}^n)\; ),$$

$$u\_\{i-1/2\}^\{n+1/2\}=\; \backslash frac\{1\}\{2\}(u\_\{i\}^n\; +\; u\_\{i-1\}^n)\; -\; \backslash frac\{\backslash Delta\; t\}\{2\backslash ,\backslash Delta\; x\}(\; f(u\_\{i\}^n)\; -\; f(u\_\{i-1\}^n)\; ).$$

Second step:

$$u\_i^\{n+1\}\; =\; u\_i^n\; -\; \backslash frac\{\backslash Delta\; t\}\{\backslash Delta\; x\}\; \backslash left[\; f(u\_\{i+1/2\}^\{n+1/2\})\; -\; f(u\_\{i-1/2\}^\{n+1/2\})\; \backslash right].$$

{{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:

$$u\_\{i\}^\{*\}=\; u\_\{i\}^n\; -\; \backslash frac\{\backslash Delta\; t\}\{\backslash Delta\; x\}(\; f(u\_\{i+1\}^n)\; -\; f(u\_\{i\}^n)\; ).$$

Second step:

$$u\_i^\{n+1\}\; =\; \backslash frac\{1\}\{2\}\; (u\_\{i\}^n\; +\; u\_\{i\}^*)\; -\; \backslash frac\{\backslash Delta\; t\}\{2\; \backslash Delta\; x\}\; \backslash left[\; f(u\_\{i\}^\{*\})\; -\; f(u\_\{i-1\}^\{*\})\; \backslash right].$$

Alternatively,

First step:

$$u\_\{i\}^\{*\}\; =\; u\_\{i\}^n\; -\; \backslash frac\{\backslash Delta\; t\}\{\backslash Delta\; x\}(\; f(u\_\{i\}^n)\; -\; f(u\_\{i-1\}^n)\; ).$$

Second step:

$$u\_i^\{n+1\}\; =\; \backslash frac\{1\}\{2\}\; (u\_\{i\}^n\; +\; u\_\{i\}^*)\; -\; \backslash frac\{\backslash Delta\; t\}\{2\; \backslash Delta\; x\}\; \backslash left[\; f(u\_\{i+1\}^\{*\})\; -\; f(u\_\{i\}^\{*\})\; \backslash right].$$

{{Reflist}}

• 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}}

{{Numerical PDE}}

{{DEFAULTSORT:Lax-Wendroff Method}}

Category:Numerical differential equations

Category:Computational fluid dynamics