Numerical Laplace transformation methods for integrating linear parabolic partial differential equations

[摘要] ENGLISH ABSTRACT: In recent years the Laplace inversion method has emerged as a viable alternativemethod for the numerical solution of PDEs. Effective methods for thenumerical inversion are based on the approximation of the Bromwich integral.In this thesis, a numerical study is undertaken to compare the efficiency ofthe Laplace inversion method with more conventional time integrator methods.Particularly, we consider the method-of-lines based on MATLAB's ODE15sand the Crank-Nicolson method.Our studies include an introductory chapter on the Laplace inversion method.Then we proceed with spectral methods for the space discretization where weintroduce the interpolation polynomial and the concept of a differentiationmatrix to approximate derivatives of a function. Next, formulas of the numericaldifferentiation formulas (NDFs) implemented in ODE15s, as well as thewell-known second order Crank-Nicolson method, are derived. In the Laplacemethod, to compute the Bromwich integral, we use the trapezoidal rule overa hyperbolic contour. Enhancement to the computational efficiency of thesemethods include the LU as well as the Hessenberg decompositions.In order to compare the three methods, we consider two criteria: Thenumber of linear system solves per unit of accuracy and the CPU time perunit of accuracy. The numerical results demonstrate that the new method,i.e., the Laplace inversion method, is accurate to an exponential order of convergencecompared to the linear convergence rate of the ODE15s and theCrank-Nicolson methods. This exponential convergence leads to high accuracywith only a few linear system solves. Similarly, in terms of computational cost, the Laplace inversion method is more efficient than ODE15s and theCrank-Nicolson method as the results show.Finally, we apply with satisfactory results the inversion method to the axialdispersion model and the heat equation in two dimensions.

[发布机构] Stellenbosch University