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Among other contributions, he is lead developer of the open source software project Clawpack for solving hyperbolic partial differential equations using the finite volume method. With Zhilin Li, he has also devised a numerical technique called the immersed interface method for solving problems with elastic boundaries or surface tension. [2] [3]
To use a finite difference method to approximate the solution to a problem, one must first discretize the problem's domain. This is usually done by dividing the domain into a uniform grid (see image). This means that finite-difference methods produce sets of discrete numerical approximations to the derivative, often in a "time-stepping" manner.
A finite difference is a mathematical expression of the form f (x + b) − f (x + a).If a finite difference is divided by b − a, one gets a difference quotient.The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
The CPU time to solve the system of equations differs substantially from method to method. Finite differences are usually the cheapest on a per grid point basis followed by the finite element method and spectral method. However, a per grid point basis comparison is a little like comparing apple and oranges.
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.
Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag. ISBN 3-540-65966-8. Randall J. LeVeque: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge 2002, ISBN 0-521-00924-3 (Cambridge Texts in Applied Mathematics
Backward finite difference [ edit ] To get the coefficients of the backward approximations from those of the forward ones, give all odd derivatives listed in the table in the previous section the opposite sign, whereas for even derivatives the signs stay the same.
LeVeque, Randall (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, ISBN 0-521-00924-3 Toro, E.F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics (2nd ed.), Springer-Verlag, ISBN 3-540-65966-8