FTCS scheme

In numerical analysis, the FTCS (Forward-Time Central-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 in time, and is conditionally stable when applied to the heat equation. When used as a method for advection equations, or more generally hyperbolic partial differential equation, it is unstable unless artificial viscosity is included. The abbreviation FTCS was first used by Patrick Roache. In numerical analysis, the FTCS (Forward-Time Central-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 in time, and is conditionally stable when applied to the heat equation. When used as a method for advection equations, or more generally hyperbolic partial differential equation, it is unstable unless artificial viscosity is included. The abbreviation FTCS was first used by Patrick Roache. The FTCS method is based on central difference in space and the forward Euler method in time, giving first-order convergence in time and second-order convergence in space. For example, in one dimension, if the partial differential equation is then, letting u ( i Δ x , n Δ t ) = u i n {displaystyle u(i,Delta x,n,Delta t)=u_{i}^{n},} , the forward Euler method is given by: The function F {displaystyle F} must be discretized spatially with a central difference scheme. This is an explicit method which means that, u i n + 1 {displaystyle u_{i}^{n+1}} can be explicitly computed (no need of solving a system of algebraic equations) if values of u {displaystyle u} at previous time level ( n ) {displaystyle (n)} are known. FTCS method is computationally inexpensive since the method is explicit. The FTCS method is often applied to diffusion problems. As an example, for 1D heat equation,