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The Lax–Friedrichs method, named after
Peter Lax Peter David Lax (1 May 1926 – 16 May 2025) was a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics. Lax made important contributions to integrable systems, fluid dynamics an ...
and Kurt O. Friedrichs, is a numerical method for the solution of
hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n - 1 derivatives. More precisely, the Cauchy problem can ...
s based on
finite difference A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly d ...
s. 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 A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data which has a single discontinuity in the domain of interest. The Riemann prob ...
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 u(x,t) of the form: u_t + a u_x = 0 on the domain b \leq x \leq c,\; 0 \leq t \leq d with initial condition u(x,0) = u_0(x)\, and the boundary conditions \begin u(b,t) &= u_b(t) \\ u(c,t) &= u_c(t). \end If one discretizes the domain (b, c) \times (0, d) to a grid with equally spaced points with a spacing of \Delta x in the x-direction and \Delta t in the t-direction, we introduce an approximation \tilde u of u u_i^n = \tilde u(x_i, t^n) ~~\text~~ \beginx_i = b + i\,\Delta x ,\\ t^n = n\,\Delta t\end ~~\text~~ \begini = 0,\ldots,N ,\\ n = 0,\ldots,M,\end where N = \frac ,\, M = \frac are integers representing the number of grid intervals. Then the Lax–Friedrichs method to approximate the partial differential equation is given by: \frac + a\frac = 0 Or, rewriting this to solve for the unknown u_i^, u_i^ = \frac \left(u_^n + u_^n\right) - a\frac \left(u_^n - u_^n\right) Where the initial values and boundary nodes are taken from \begin u_i^0 &= u_0(x_i) \\ u_0^n &= u_b(t^n) \\ u_N^n &= u_c(t^n). \end


Extensions to Nonlinear Problems

A nonlinear hyperbolic conservation law is defined through a flux function f : u_t + ( f(u) )_x = 0. In the case of f(u) = a u , we end up with a scalar linear problem. Note that in general, u is a vector with m equations in it. The generalization of the Lax-Friedrichs method to nonlinear systems takes the form u_i^ = \frac \left(u_^n + u_^n\right) - \frac \left( f( u_^n ) - f( u_^n ) \right). 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: u_i^ = u^n_i - \frac \left( \hat^n_ - \hat^n_ \right), where \hat^n_ = \frac \left( f_ + f_ \right) - \frac \left( u^n_ - u^n_ \right). Without the extra terms u^n_i and u^n_ in the discrete flux, \hat^n_ , one ends up with the
FTCS scheme In numerical analysis, the FTCS (forward time-centered space) method is a finite difference method used for numerically solving the heat equation and similar parabolic partial differential equations. It is a first-order method in time, explicit i ...
, which is well known to be unconditionally unstable for hyperbolic problems.


Stability and accuracy

This method is
explicit Explicit refers to something that is specific, clear, or detailed. It can also mean: * Explicit knowledge, knowledge that can be readily articulated, codified and transmitted to others * Explicit (text), the final words of a text; contrast with inc ...
and first order accurate in time and first order accurate in space ( O(\Delta t) + O(/)) provided u_0(x),\, u_b(t),\, u_c(t) are sufficiently-smooth functions. Under these conditions, the method is
stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...
if and only if the following condition is satisfied: \left, a\frac \ \leq 1. (A
von Neumann stability analysis In numerical analysis, von Neumann stability analysis (also known as Fourier stability analysis) is a procedure used to check the stability of finite difference schemes as applied to linear partial differential equations. The analysis is based on ...
can show the necessity of this stability condition.) The Lax–Friedrichs method is classified as having second-order
dissipation In thermodynamics, dissipation is the result of an irreversible process that affects a thermodynamic system. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to a final form, wh ...
and third order
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns * Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variat ...
. For functions that have discontinuities, the scheme displays strong dissipation and dispersion; see figures at right.


References

* * {{DEFAULTSORT:Lax-Friedrichs method Numerical differential equations Computational fluid dynamics