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In numerical analysis, finite-difference methods ( FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time domain (if applicable) are discretized, or broken into a finite number of intervals, and the values of the solution at the end ...
In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. [1] It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable.
Heat equation. Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. The height and redness indicate the temperature at each point. The initial state has a uniformly hot hoof-shaped region (red) surrounded by uniformly cold region (yellow). As time passes the heat diffuses into the cold region.
Example: 2D diffusion equation Stencil figure for the alternating direction implicit method in finite difference equations. The traditional method for solving the heat conduction equation numerically is the Crank–Nicolson method. This method results in a very complicated set of equations in multiple dimensions, which are costly to solve.
Equation gives the stability requirement for the FTCS scheme as applied to one-dimensional heat equation. It says that for a given , the allowed value of must be small enough to satisfy equation . Similar analysis shows that a FTCS scheme for linear advection is unconditionally unstable.
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. [1] It is a first-order method in time, explicit in time, and is conditionally stable when applied to the heat equation.
In this method, the basic shape function is modified to obtain the upwinding effect. This method is an extension of Runge–Kutta discontinuous for a convection-diffusion equation. For time-dependent equations, a different kind of approach is followed. The finite difference scheme has an equivalent in the finite element method (Galerkin method ...
Courant–Friedrichs–Lewy condition. In mathematics, the convergence condition by Courant–Friedrichs–Lewy is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of explicit time integration schemes, when these are used for the ...