Suppose we want to solve a hyperbolic conservation law $u_t+f(u)_x=0$. I really like to use Lax-Wendroff, which reads

$u_j^{n+1} = u_j^n -\frac{\Delta t}{\Delta x}(g(u_{j+1}^n,u_j^n)-g(u_j^n,u_{j-1}^n))$

where

$g(u,w) = \frac12(f(v)+f(w)) - \frac{\Delta t}{2\Delta x}\vert\frac{f(w)-f(v)}{w-v}\vert^2(w-v)$

or for $f(u)=au$ (linear advection),

$g(v,w)=\frac12a((v+w)-\frac{\Delta t}{\Delta x}a(w-v))$.

My question: Is there anything similar that has a higher order? I'm not talking about those fancy high resolution, (W)ENO, MUSCL,... schemes - just a plain third or fourth order, stable scheme that works for arbitrary $f$.

Suppose we want to solve a hyperbolic conservation law $u_t+f(u)_x=0$. I really like to use Lax-Wendroff, which reads

$u_j^{n+1} = u_j^n -\frac{\Delta t}{\Delta x}(g(u_{j+1}^n,u_j^n)-g(u_j^n,u_{j-1}^n))$

where

$g(v,w) = \frac12(f(v)+f(w)) - \frac{\Delta t}{2\Delta x}\vert\frac{f(w)-f(v)}{w-v}\vert^2(w-v)$

or for $f(u)=au$ (linear advection),

$g(v,w)=\frac12a((v+w)-\frac{\Delta t}{\Delta x}a(w-v))$.

My question: Is there anything similar that has a higher order? I'm not talking about those fancy high resolution, (W)ENO, MUSCL,... schemes - just a plain third or fourth order, stable scheme that works for arbitrary $f$.

Suppose we want to solve a hyperbolic conservation law $u_t+f(u)_x=0$. I really like to use Lax-Wendroff in cases I don't need a monotone method, which reads

$u_j^{n+1} = u_j^n -\frac{\Delta t}{\Delta x}(g(u_{j+1}^n,u_j^n)-g(u_j^n,u_{j-1}^n))$

where

$g(u,w) = \frac12(f(v)+f(w)) - \frac{\Delta t}{2\Delta x}\vert\frac{f(w)-f(v)}{w-v}\vert^2(w-v)$

or for $f(u)=au$ (linear advection),

$g(v,w)=\frac12a((v+w)-\frac{\Delta t}{\Delta x}a(w-v))$.

My question: Is there anything similar that has a higher order? I'm not talking about those fancy high resolution, (W)ENO, MUSCL,... schemes - just a plain third or fourth order, stable scheme that works for arbitrary $f$.

Suppose we want to solve a hyperbolic conservation law $u_t+f(u)_x=0$. I really like to use Lax-Wendroff, which reads

$u_j^{n+1} = u_j^n -\frac{\Delta t}{\Delta x}(g(u_{j+1}^n,u_j^n)-g(u_j^n,u_{j-1}^n))$

where

$g(u,w) = \frac12(f(v)+f(w)) - \frac{\Delta t}{2\Delta x}\vert\frac{f(w)-f(v)}{w-v}\vert^2(w-v)$

or for $f(u)=au$ (linear advection),

$g(v,w)=\frac12a((v+w)-\frac{\Delta t}{\Delta x}a(w-v))$.

My question: Is there anything similar that has a higher order? I'm not talking about those fancy high resolution, (W)ENO, MUSCL,... schemes - just a plain third or fourth order, stable scheme that works for arbitrary $f$.