TheLax–Friedrichs method, named afterPeter Lax andKurt O. Friedrichs, is anumerical method for the solution ofhyperbolic partial differential equations based onfinite differences. The method can be described as theFTCS (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 toGodunov's scheme, where one avoids solving aRiemann problem at each cell interface, at the expense of adding artificial viscosity.

Consider a one-dimensional, linear hyperbolic partial differential equation for${\displaystyle u(x,t)}$ of the form:$${\displaystyle u_t+a{u}_x=0}$$on the domain$${\displaystyle b\le x\le c,0\le t\le d}$$with initial condition$${\displaystyle u(x,0)=u_0(x)}$$and the boundary conditions

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

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

Where the initial values and boundary nodes are taken from

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

In the case of${\displaystyle f(u)=au}$, we end up with a scalar linear problem. Note that in general,${\displaystyle u}$ is a vector with${\displaystyle m}$ equations in it.The generalization of the Lax-Friedrichs method to nonlinear systems takes the form^[1]$${\displaystyle u_i^{n+1}=\frac{1}{2}\left(u_{i+1}^n+u_{i-1}^n\right)-\frac{\mathrm{\Delta }t}{2\mathrm{\Delta }x}\left(f(u_{i+1}^n)-f(u_{i-1}^n)\right).}$$

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

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

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

(Avon Neumann stability analysis can show the necessity of this stability condition.) The Lax–Friedrichs method is classified as having second-orderdissipation and third orderdispersion.^[2] For functions that havediscontinuities, the scheme displays strong dissipation and dispersion;^[3] see figures at right.

• 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

• Numerical differential equations
• Computational fluid dynamics

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