Lax–Friedrichs method

Template:Short description The Lax–Friedrichs method, named after Peter Lax and Kurt O. Friedrichs, is a numerical method for the solution of hyperbolic partial differential equations based on finite differences. The method can be described as the FTCS (forward in time, centered in space) scheme with a numerical dissipation term of 1/2. One can view the Lax–Friedrichs method as an alternative to Godunov's scheme, where one avoids solving a Riemann problem at each cell interface, at the expense of adding artificial viscosity.

Illustration for a Linear Problem

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 $$b\le x\le c,0\le t\le d$$ with initial condition $$u(x,0)=u_0(x)$$ and the boundary conditions $$\begin{array}{cc}u(b,t)& =u_b(t)\\ u(c,t)& =u_c(t).\end{array}$$

If one discretizes the domain $(b,c)\times (0,d)$ to a grid with equally spaced points with a spacing of $\mathrm{\Delta }x$ in the $x$-direction and $\mathrm{\Delta }t$ in the $t$-direction, we introduce an approximation $\tilde{u}$ of $u$ $$u_i^n=\tilde{u}(x_i,t^n)\text{ with }\begin{array}{c}{x}_i=b+i\mathrm{\Delta }x,\\ t^n=n\mathrm{\Delta }t\end{array}\text{ for }\begin{array}{c}i=0,\dots ,N,\\ n=0,\dots ,M,\end{array}$$ where $$N=\frac{c-b}{\mathrm{\Delta }x},M=\frac{d}{\mathrm{\Delta }t}$$ are integers representing the number of grid intervals. Then the Lax–Friedrichs method to approximate the partial differential equation is given by: $$\frac{u_i^{n+1}-\frac{1}{2}(u_{i+1}^n+u_{i-1}^n)}{\mathrm{\Delta }t}+a\frac{u_{i+1}^n-u_{i-1}^n}{2\mathrm{\Delta }x}=0$$

Or, rewriting this to solve for the unknown $u_i^{n+1},$ $$u_i^{n+1}=\frac{1}{2}(u_{i+1}^n+u_{i-1}^n)-a\frac{\mathrm{\Delta }t}{2\mathrm{\Delta }x}(u_{i+1}^n-u_{i-1}^n)$$

Where the initial values and boundary nodes are taken from $$\begin{array}{cc}{u}_i^0& =u_0(x_i)\\ u_0^n& =u_b(t^n)\\ u_N^n& =u_c(t^n).\end{array}$$

Extensions to Nonlinear Problems

A nonlinear hyperbolic conservation law is defined through a flux function $f$: $$u_t+(f(u){)}_x=0.$$

In the case of $f(u)=au$, 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 $$u_i^{n+1}=\frac{1}{2}(u_{i+1}^n+u_{i-1}^n)-\frac{\mathrm{\Delta }t}{2\mathrm{\Delta }x}(f(u_{i+1}^n)-f(u_{i-1}^n)).$$

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: $$u_i^{n+1}=u_i^n-\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}({\hat{f}}_{i+1/2}^n-{\hat{f}}_{i-1/2}^n),$$ where $${\hat{f}}_{i-1/2}^n=\frac{1}{2}(f_{i-1}+f_i)-\frac{\mathrm{\Delta }x}{2\mathrm{\Delta }t}(u_i^n-u_{i-1}^n).$$

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

Stability and accuracy

File:LF-Initial.png

Example problem initial condition

File:LF-Solution.png

Lax-Friedrichs solution

This method is explicit and first order accurate in time and first order accurate in space ($O(\mathrm{\Delta }t)+O(\mathrm{\Delta }{x}^2/\mathrm{\Delta }t))$ provided $u_0(x),u_b(t),u_c(t)$ are sufficiently-smooth functions. Under these conditions, the method is stable if and only if the following condition is satisfied: $$\left|a\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}\right|\le 1.$$

(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. For functions that have discontinuities, the scheme displays strong dissipation and dispersion; see figures at right.

References

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