Lax–Friedrichs method

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 of the form:

on the domain

with initial condition

and the boundary conditions

If one discretizes the domain to a grid with equally spaced points with a spacing of in the -direction and in the -direction, we define

where

are integers representing the number of grid intervals. Then the Lax–Friedrichs method for solving the above partial differential equation is given by:

Or, rewriting this to solve for the unknown

Where the initial values and boundary nodes are taken from

Extensions to Nonlinear Problems

A nonlinear hyperbolic conservation law is defined through a flux function :

In the case of , we end up with a scalar linear problem. Note that in general, is a vector with equations in it. The generalization of the Lax-Friedrichs method to nonlinear systems takes the form[1]

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:

where

Without the extra terms and in the discrete flux, , one ends up with the FTCS scheme, which is well known to be unconditionally unstable for hyperbolic problems.

Stability and accuracy

Example problem initial condition
Lax-Friedrichs solution

This method is explicit and first order accurate in time and first order accurate in space ( provided are sufficiently-smooth functions. Under these conditions, the method is stable if and only if the following condition is satisfied:

(A von Neumann stability analysis can show the necessity of this stability condition.) The Lax–Friedrichs method is classified as having second-order dissipation and third order dispersion (Chu 1978, pg. 304). For functions that have discontinuities, the scheme displays strong dissipation and dispersion (Thomas 1995, §7.8); see figures at right.

References

  1. LeVeque, Randall J. Numerical Methods for Conservation Laws", Birkhauser Verlag, 1992, p. 125.
  • DuChateau, Paul; Zachmann, David (2002), Applied Partial Differential Equations, New York: Dover Publications, ISBN 978-0-486-41976-3.
  • Thomas, J. W. (1995), Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, 22, Berlin, New York: Springer-Verlag, ISBN 978-0-387-97999-1.
  • Chu, C. K. (1978), Numerical Methods in Fluid Mechanics, Advances in Applied Mechanics, 18, New York: Academic Press, ISBN 978-0-12-002018-8.
  • Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 10.1.2. Lax Method", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
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