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 ) {\displaystyle u(x,t)} of the form:
u t + a u x = 0 {\displaystyle u_{t}+au_{x}=0}
on the domain
b ≤ x ≤ c , 0 ≤ t ≤ d {\displaystyle b\leq x\leq c,\;0\leq t\leq d}
with initial condition
u ( x , 0 ) = u 0 ( x ) {\displaystyle u(x,0)=u_{0}(x)\,}
and the boundary conditions
u ( b , t ) = u b ( t ) u ( c , t ) = u c ( t ) . {\displaystyle {\begin{aligned}u(b,t)&=u_{b}(t)\\u(c,t)&=u_{c}(t).\end{aligned}}}
If one discretizes the domain ( b , c ) × ( 0 , d ) {\displaystyle (b,c)\times (0,d)} to a grid with equally spaced points with a spacing of Δ x {\displaystyle \Delta x} in the x {\displaystyle x} -direction and Δ t {\displaystyle \Delta t} in the t {\displaystyle t} -direction, we introduce an approximation u ~ {\displaystyle {\tilde {u}}} of u {\displaystyle u}
u i n = u ~ ( x i , t n ) with x i = b + i Δ x , t n = n Δ t for i = 0 , … , N , n = 0 , … , M , {\displaystyle 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 = c − b Δ x , M = d Δ t {\displaystyle 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:
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