Study On Semi-discrete Finite Difference Methods For The Heat Equation

Heat equation is of its own value in describing many natural phenomena such as the diffusion of polluted substance in water, air and salinity in the ocean etc. Therefore the computation of the heat equations has drawn much attention of many researchers . This paper studies semi-discrete finite difference methods of constant coefficient heat equations . at present the methods of solving heat equations are finite difference method(FDM), finite element method, and some newly developing methods, such as finite volume method, spectral method etc . However FDM is still one of the useful, efficient methods for solving linear heat equations . There are explicit and implicit schemes in FDM . explicit scheme are easy to compute and make programmes, but it has a problem of conditional stability and the exactness is not high . Even if crank-nicolson implicit scheme is unconditionally stable, but it's volume of calculations is large and it demands to solve large system of linear equations, which has a large coefficient matrix, in some conditions coefficient matrix is ill-conditioned . So we have studied semi-discrete FDM of heat equations . In part one, we firstly applied the central difference formula for the second derivative to the heat equations on variable x, time variable t remain continuous the equation transformed into a large series of ODES . and discussed consistency, convergence, stability . forward Euler, backward Euler, R-K methods are applied to ODES, and we discussed the existing problems and solving processes and methods and Through doing numerical experiment (1). A better result is obtained . In part two we discussed the study of semi-discrete FDM one dimensional contained low order item heat equations consistency, convergence , stability are analysis and make numerical experiment(2), which is related to it, and achieved a better result and the problems of stiffness . In part three, we extended the conclusion of part(2) to two dimensional contained low order item heat equation and discussed its consistency, convergence, stability and we obtained easily calculated unconditionally stable explicitscheme of solving heat equation, moreover we did numerical experiment(3). With our numerical experiments, we proved our method is applicable, useful, efficient methods for solving linear heat equations .