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
[edit]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 introduce an approximation of where are integers representing the number of grid intervals. Then the Lax–Friedrichs method to approximate the 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
[edit]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
[edit]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.[2] For functions that have discontinuities, the scheme displays strong dissipation and dispersion;[3] see figures at right.
References
[edit]- ^ LeVeque, Randall J. (1992). Numerical methods for conservation laws. Basel: Birkhäuser Verlag. p. 125. ISBN 978-3-0348-8629-1. OCLC 828775522.
- ^ Chu, C. K. (1978), Numerical Methods in Fluid Mechanics, Advances in Applied Mechanics, vol. 18, New York: Academic Press, p. 304, ISBN 978-0-12-002018-8
- ^ Thomas, J. W. (1995), Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, vol. 22, Berlin, New York: Springer-Verlag, §7.8, ISBN 978-0-387-97999-1
- DuChateau, Paul; Zachmann, David (2002), Applied Partial Differential Equations, New York: Dover Publications, ISBN 978-0-486-41976-3
- Press, William H; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 20.1.2. Lax Method", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8