Lax–Friedrichs method

Consider a one-dimensional, linear hyperbolic partial differential equation for $u(x,t)$ of the form:

$$u\_t\; +\; a\; u\_x\; =\; 0$$

on the domain

$$b\; \backslash leq\; x\; \backslash leq\; c,\backslash ;\; 0\; \backslash leq\; t\; \backslash leq\; d$$

with initial condition

$$u(x,0)\; =\; u\_0(x)\backslash ,$$

and the boundary conditions

$$\backslash begin\{align\}$$

u(b,t) &= u_b(t) \\

u(c,t) &= u_c(t).

\end{align}

If one discretizes the domain $(b,\; c)\; \backslash times\; (0,\; d)$ to a grid with equally spaced points with a spacing of $\backslash Delta\; x$ in the $x$-direction and $\backslash Delta\; t$ in the $t$-direction, we introduce an approximation $\backslash tilde\; u$ of $u$

$$u\_i^n\; =\; \backslash tilde\; u(x\_i,\; t^n)\; ~~\backslash text\{\; with\; \}~~$$

\begin{array}{l}x_i = b + i\,\Delta x ,\\ t^n = n\,\Delta t\end{array}

~~\text{ for }~~

\begin{array}{l}i = 0,\ldots,N ,\\ n = 0,\ldots,M,\end{array}

where

$$N\; =\; \backslash frac\{c\; -\; b\}\{\backslash Delta\; x\}\; ,\backslash ,\; M\; =\; \backslash frac\{d\}\{\backslash Delta\; t\}$$

are integers representing the number of grid intervals. Then the Lax–Friedrichs method to approximate the partial differential equation is given by:

$$\backslash frac\{u\_i^\{n+1\}\; -\; \backslash frac\{1\}\{2\}(u\_\{i+1\}^n\; +\; u\_\{i-1\}^n)\}\{\backslash Delta\; t\}\; +\; a\backslash frac\{u\_\{i+1\}^n\; -\; u\_\{i-1\}^n\}\{2\backslash ,\backslash Delta\; x\}\; =\; 0$$

Or, rewriting this to solve for the unknown $u\_i^\{n+1\},$

$$u\_i^\{n+1\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left(u\_\{i+1\}^n\; +\; u\_\{i-1\}^n\backslash right)\; -\; a\backslash frac\{\backslash Delta\; t\}\{2\backslash ,\backslash Delta\; x\}\; \backslash left(u\_\{i+1\}^n\; -\; u\_\{i-1\}^n\backslash right)$$

Where the initial values and boundary nodes are taken from

$$\backslash begin\{align\}$$

u_i^0 &= u_0(x_i) \\

u_0^n &= u_b(t^n) \\

u_N^n &= u_c(t^n).

\end{align}

A nonlinear hyperbolic conservation law is defined through a flux function $f$:

$$u\_t\; +\; (\; f(u)\; )\_x\; =\; 0.$$

In the case of $f(u)\; =\; a\; u$, we end up with a scalar linear problem. Note that in general, $u$ is a vector with $m$ equations in it.

The generalization of the Lax-Friedrichs method to nonlinear systems takes the form{{Cite book |last=LeVeque |first=Randall J. |url=https://www.worldcat.org/oclc/828775522 |title=Numerical methods for conservation laws |date=1992 |publisher=Birkhäuser Verlag |isbn=978-3-0348-8629-1 | page=125 |location=Basel |oclc=828775522}}

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

This method is conservative and first order accurate, hence quite dissipative. It can, however be used as a building block for building high-order numerical schemes for solving hyperbolic partial differential equations, much like Euler time steps can be used as a building block for creating high-order numerical integrators for ordinary differential equations.

We note that this method can be written in conservation form:

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

where

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

Without the extra terms $u^n\_i$ and $u^n\_\{i-1\}$ in the discrete flux, $\backslash hat\{f\}^n\_\{i-1/2\}$, one ends up with the FTCS scheme, which is well known to be unconditionally unstable for hyperbolic problems.

File:LF-Initial.png

File:LF-Solution.png

This method is explicit and first order accurate in time and first order accurate in space ($O(\backslash Delta\; t)\; +\; O(\{\backslash Delta\; x^2\}/\{\backslash Delta\; t\}))$ provided $u\_0(x),\backslash ,\; u\_b(t),\backslash ,\; u\_c(t)$ are sufficiently-smooth functions. Under these conditions, the method is stable if and only if the following condition is satisfied:

$$\backslash left|\; a\backslash frac\{\backslash Delta\; t\}\{\backslash Delta\; x\}\; \backslash right|\; \backslash leq\; 1.$$

(A von Neumann stability analysis can show the necessity of this stability condition.) The Lax–Friedrichs method is classified as having second-order dissipation and third order dispersion.{{Citation | last1=Chu| first1=C. K. | title=Numerical Methods in Fluid Mechanics | publisher=Academic Press | location=New York | series=Advances in Applied Mechanics | isbn=978-0-12-002018-8 | year=1978 | volume=18 | page=304}} For functions that have discontinuities, the scheme displays strong dissipation and dispersion;{{Citation | last1=Thomas | first1=J. W. | title=Numerical Partial Differential Equations: Finite Difference Methods | publisher=Springer-Verlag | location=Berlin, New York | series=Texts in Applied Mathematics | isbn=978-0-387-97999-1 | year=1995 | volume=22 | at=§7.8}} see figures at right.

{{reflist}}

• {{Citation | last1=DuChateau | first1=Paul | last2=Zachmann | first2=David | title=Applied Partial Differential Equations | publisher=Dover Publications | location=New York | isbn=978-0-486-41976-3 | year=2002 | url-access=registration | url=https://archive.org/details/appliedpartialdi00duch }}
• {{Citation |last1=Press |first1=William H |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 20.1.2. Lax Method |chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1034}}

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Category:Numerical differential equations

Category:Computational fluid dynamics