MSc. (Maths: NA & Comp. Sc.)
 A Comparative Study of Some of the Most Common Numerical Methods for Solving  Parabolic Partial Differential Equations
 
Denis Ssebuggwawo
 
Master of Science (Maths: NA & Comp. Sc.)  – 1997
 
Department of Mathematics (Math),  School of Physical Sciences (SPS)
College  of Natural Sciences (CNS),  Makerere University (MAK), Kampala – (Uganda)
 
Supervisor: Mr. E.M. Kizza (MAK) R.I.P.
Internal Examiner: Prof. P.J.M. Mangheni (MAKUCU),
External Examiner: Prof. Dr. Ralph W.P. Masenge (UDSOUT)

Extended Abstract

The objective of this research was two-fold:
    1. We  tried to carry out a comparative study of some of the most common numerical methods for solving a parabolic partial differential equation. The comparison was done with respect to:   
 
  • Accuracy and error reduction
        The investigation here was done mainly to find out:
        (i)  the convergence of a numerical method
        (ii)  the stability and consistency of the numerical method
 
  • Storage and computing time
        Here we analysed:
        (i)  the nature of the matrices produced by solving the resulting algebraic equations for each of the numerical methods
        (ii)  the ease of programming a numerical method
  • Space finite elements and space-time finite elements.
        An answer to the following question was sought:
        Which of the two types of elements above is more suitable for solving a parabolic partial differential equation?
 
  • The popularity of a numerical method for solving a parabolic partial differential equation  
      2. The second objective of this research was to investigate how accuracy may be analysed
  • With recourse to an analytical (exact) solution  
  • Without recourse to the analytical (exact) solution
        This analysis was done with respect to:
        (i)  refinement of the mesh without altering the order of accuracy of a numerical method
        (ii)  the use of higher-order approximation on a fixed grid.

We may conclude from our research that we succesfully and explicitly identified:

  • the best and most popular numerical method for solving a parabolic partial differential equation.
  • the most efficient finite element and
  • the best technique for improving accuracy and reducing the error.

Though this claim may seem to be supported by some of the results in this manuscript, we believe that this would be making a mistake of some pernicious sort, evidencing some lack of valuable information about a parabolic partial differential equation, and the numerical methods. From the computational perspective, we believe the best thing to do, is to have both methods ( the finite difference and the finite element methods), since each of them has its pros and cons. This will always give the numerical analyst an alternative weapon in his/her approximation armoury so long as the fundamental computational concerns: “stability”, ” consisitency” and “convergence” of the approximation is guaranteed, and the problem to be solved is well-posed. Therefore, the choice of the method will always depend on the problem and/or its boundary conditions and the nature of the grid ( regular or irregular ). Also the choice of the technique of accuracy improvement and reduction of the error depends on the problem being solved, the behaviour of the solution, and the numerical method being used to solve a given problem.

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