Explicit-Implicit Difference Computational Methods For Solving Two Kinds Of Time Fractional Evolution Equations

The numerical algorithms' researches of the time fractional evolution equations have become one of the main topics in anomalous diffusion phenomenons'researches.In this dissertation,three kinds of difference schemes are constructed for the time fractional sub-diffusion equation and the time fractional convection-diffusion equation:explicit-implicit serial difference scheme(E-I)and implicit-explicit serial difference scheme(I-E),pure alternative segment explicit-implicit(PASE-I)and pure alternative segment implicit-explicit(PASI-E)parallel difference schemes,alternative segment explicit-implicit(ASE-I)and alternative segment implicit-explicit(ASI-E)parallel difference schemes.First we construct E-I and I-E schemes,which are based on the combination of the explicit scheme and implicit scheme.Theoretical analyses have shown that the solution of E-I(I-E)scheme is uniquely solvable.At the same time the stability and convergence of the scheme are proved by the Fourier method.Numerical experiments verify the theoretical analyses under the fixed number of grid points and show the computational efficiency of E-I(I-E)scheme is 45%higher than the implicit scheme under the premise of unconditional stability and having better accuracy.Therefore it is feasible to use the scheme for solving the time fractional diffusion equation.Secondly,combining with the alternating segment technique,a kind of parallel numerical method with parallel nature which are the pure alternative segment explicit-implicit difference method and pure alternative segment implicit-explicit difference method is proposed for solving the time fractional diffusion equation based on the idea of E-I and I-E serial method.Theoretical analyses have shown that the solutions of PASE-I and PASI-E scheme are uniquely solvable.At the same time the stability and convergence of the schemes are proved by the Fourier method and the mathematical induction.Numerical experiments verify the theoretical analyses which show that the convergence rates are spatially second-order and temporally 2-? order.Meanwhile the PASE-I and PASI-E schemes have obvious parallel properties because of the higher computational efficiency.Finally we construct ASE-I and ASI-E scheme through adding the Saul ' yev asymmetric format to the "inner boundary point" based on the PASE-I scheme and PASI-E scheme.Compared to the PASE-I scheme,it has a faster computational speed under the guarantee of the convergence and accuracy.At the same time,numerical experiments were compared and analyzed for the E-I scheme,the PASE-I scheme and the ASE-I scheme in solving the time fractional sub-diffusion equation.The computation accuracy of the three schemes is better,and the computational time of ASE-I scheme is relatively short.Meanwhile the three difference schemes are applied to the solution of the time fractional convection diffusion equation,which have a good practical application value.