Lax–Friedrichs method

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: $$u_{t}+au_{x}=0$$ on the domain $$b\leq x\leq c,\;0\leq t\leq d$$ with initial condition $$u(x,0)=u_{0}(x)\,$$ and the boundary conditions $${\begin{aligned}u(b,t)&=u_{b}(t)\\u(c,t)&=u_{c}(t).\end{aligned}}$$

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

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

Where the initial values and boundary nodes are taken from $${\begin{aligned}u_{i}^{0}&=u_{0}(x_{i})\\u_{0}^{n}&=u_{b}(t^{n})\\u_{N}^{n}&=u_{c}(t^{n}).\end{aligned}}$$

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 $$u_{i}^{n+1}={\frac {1}{2}}\left(u_{i+1}^{n}+u_{i-1}^{n}\right)-{\frac {\Delta t}{2\,\Delta x}}\left(f(u_{i+1}^{n})-f(u_{i-1}^{n})\right).$$

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

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

Example problem initial condition

Lax-Friedrichs solution

This method is explicit and first order accurate in time and first order accurate in space ($O(\Delta t)+O({\Delta x^{2}}/{\Delta t}))$ 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: $$\left|a{\frac {\Delta t}{\Delta x}}\right|\leq 1.$$

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

References

• DuChateau, Paul; Zachmann, David (2002), Applied Partial Differential Equations, New York: Dover Publications, ISBN 978-0-486-41976-3
• Press, William H; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 20.1.2. Lax Method", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8