Lax–Friedrichs method

Consider a one-dimensional, linear hyperbolic partial differential equation for ${\displaystyle u(x,t)}$ of the form: $${\displaystyle u_{t}+au_{x}=0}$$ on the domain $${\displaystyle b\leq x\leq c,\;0\leq t\leq d}$$ with initial condition $${\displaystyle u(x,0)=u_{0}(x)\,}$$ and the boundary conditions $${\displaystyle {\begin{aligned}u(b,t)&=u_{b}(t)\\u(c,t)&=u_{c}(t).\end{aligned}}}$$

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

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

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

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