Finite-Difference Approximations to the Heat Equation
نویسنده
چکیده
This article provides a practical overview of numerical solutions to the heat equation using the finite difference method. The forward time, centered space (FTCS), the backward time, centered space (BTCS), and Crank-Nicolson schemes are developed, and applied to a simple problem involving the one-dimensional heat equation. Complete, working Matlab codes for each scheme are presented. The results of running the codes on finer (one-dimensional) meshes, and with smaller time steps is demonstrated. These sample calculations show that the schemes realize theoretical predictions of how their truncation errors depend on mesh spacing and time step. The Matlab codes are straightforward and allow the reader to see the differences in implementation between explicit method (FTCS) and implicit methods (BTCS and Crank-Nicolson). The codes also allow the reader to experiment with the stability limit of the FTCS scheme.
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