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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 $f$.

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  • $\begingroup$ Similar in what way? And what "fancy" things disqualify an answer? $\endgroup$ – David Ketcheson Apr 20 '13 at 17:03
  • $\begingroup$ Similar in properties. You need the second term in the definition of $g$ because the first one alone, which is second order as well, yields an unstable scheme. (W)ENO, MUSCL,... all have some more tricks to handle discontinuities like flux limiters. I wanted to know if there are plain third or fourth order schemes like Lax-Wendroff. Gibbs will be there, of course, but it should be stable. $g(v_0,\dots,v_3) = \frac{7}{12}(f(v_1)+f(v_2)) - \frac{1}{12}(f(v_0) + f(v_3))$ yields a fourth order scheme, but it's not stable. $\endgroup$ – Anke Apr 20 '13 at 17:11
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There are plenty of higher order schemes out there. But as per Godunov's theorem, only first order scheme can be monotone and hence not create oscillations. This resource gives a brief idea about the construction and analysis of finite difference schemes.

In REA (Reconstruct, Evolve , Average) algorithm, required order polynomial is reconstructed and corresponding variable values are interpolated at cell face. This gives the fluxes at the cell faces. (This is same as $g(u_{j+{1}},u_{j})$ for face $j + \frac{1}{2}$ for above mentioned scheme, in which a straight line is reconstructed from cell averages).

Then cell values are updated using this fluxes.

Leveque's book "Finite Volume Methods for Hyperbolic Problems" gives detailed information on this. Depending on your choice of stencil, you can create an arbitrarily high order scheme. But there will always be oscillations near discontinuity if it is a high order scheme.

Other sources of high order schemes are,

1) DRP schemes (this paper also discusses formulation of standard FD schemes of arbitrary high order)

2) Discontinuous / continuous Galerkin methods (These can have arbitrarily high order of accuracy, but reconstruction unlike FVM, takes place within an element. Cell averaged values are not used for reconstruction)

3) Spectral methods

Some resources for the numerical shcemes,

1) "Finite Volume methods for hyperbolic problems", Randall Leveque

2) "Computational Gasdynamics", Culbert Laney (Discusses ENO, MUSCL nicely too)

3) "Riemann Solvers and Numerical Methods for Fluid Dynamics", E.F.Toro

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  • $\begingroup$ Thanks for the resource, looks like an extensive reference. As for the rest of your answer: I'm aware of the difficulties with discontinuities. Still, my question was explicitly about Lax-Wendroff type methods. Lax-Wendroff is stable, FTCS, for example, is not. Of course, you get oscillations, but still, I'm interested in this type of method. $\endgroup$ – Anke Apr 18 '13 at 13:46

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