 # Numerical solutions of a partial differential equation

Walker, Adam H (1967) Numerical solutions of a partial differential equation. MSc(R) thesis, University of Glasgow.

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## Abstract

The subject of the thesis is the numerical integration of the Partial Differential Equation: deltaf/deltat=fD delta2f/deltax2 for 0≤x≤1. In particular, the case p=1 is dealt with under the conditions: f(0,t)=c1;f(x,0)=c2 for 0≤x≤1; deltaf/deltax x=1 = 0. Also, several other cases are treated in order to bring out various aspects of the techniques used. The case p=-1 is treated in order to obtain comparisons with known analytical solutions. In the case p=0 the techniques are entirely linear and one method of integration used demands the inversion of a tri-diagonal matrix. Other values of p are chosen in order to illustrate various aspects of the stability theory and demonstrate the generality of the methods. In the first chapter, an analytic approach is adopted and the equation is classified by its non-linearity and by the fact that it is Parabolic. The equation is non-linear in the sense that it contains a product of the dependent variable and a partial derivative. These observations determine the approach to the numerical solution. In this chapter an attempt is made to find analytical solutions but it appears that such solutions are only useful for highly specialised initial and boundary conditions. Even when such solutions exist, they are usually implicit and very unwieldy. The exception to this is the case p=-1 which can be seen to have a simple analytical solution; f(x,t) = ten (pi/4+x/2+t) for suitable boundary and initial conditions. In the second chapter, three numerical methods of integration are discussed in detail. These are: 1. A simple explicit method. 2. A semi-explicit method which requires a special starting procedure. The method used is known as the DuFort-Frankel method. 3. An implicit method based on the well-known Crank-Nicolson technique. This method reduces to the solution of sets of simultaneous non-linear equations. The stability of the three methods is dealt with empirically by comparison with the linear heat-conduction equation. The results obtained may be stated briefly as follows. 1. For the explicit method we must have: fpdeltat/(deltax)2; fp≥0. 2. For the DuFort-Frankel method: fP≥0; deltat/(deltax)2 is unrestricted. 3. For the Crank-Nicolson method: fP≥0, providing that the parameter r used to combine the forward difference and backward difference representations of deltaf/deltat is greater than deltat and deltax are the step-lengths in t, x respectively. The second method is found to exhibit the phenomenon of inconsistency unless deltat/deltax is kept small. Truncation errors and the treatment of the singularity are mentioned briefly in this chapter. The main portion of the chapter is devoted to the development of the Crank-Nicolson simultaneous equations and methods for the solution of these equations. Iterative methods for the solution of the equations are treated in detail. Explicit analytical methods for the solution of the equations are ignored, since they are clumay to programme. Two iterative processes are given. One is an extension of Newtwon-Raphson iteration and the other is a generalisation of direct functional iteration. Three separate methods are investigated for the generation of first approximations for the iterative processes. These are: (a) The explicit formula mentioned earlier. (b) The Newton Backward Difference extrapolation formula. (c) Use of f (x,t-deltat) as a first approximation to f (x, t). In the third chapter, the main results are presented and discussed. It is found that the empirical stability theory given in the second chapter gives agreement with the numerical results. The DuFort-Frankel method is seen to given inconsistency for values of deltat/(deltax)2 higher than those allowed for stability in the simple explicit method. The numerical results obtained by the explicit and Crank-Nicolson methods are found to agree fairly closely. The Crank-Nicolson method gives good agreement with the analytical solution for p=-1; usually the agreement is much closer than 1%. The Newton-Raphson iterative method is found to be much superior to the direct iteration process and it is found that extremely fast convergence may be obtained by making the parameter r just greater than 1/2. At the end of the chapter, a review of the investigation is given and some conclusions of a general nature are drawn. The appendices contain a brief account of the programmes used and some of the numerical results obtained, as well as a short list of the books found useful in the solution of the problems.

Item Type: Thesis (MSc(R)) Masters Mathematics 1967 Enlighten Team glathesis:1967-72607 Copyright of this thesis is held by the author. 11 Jun 2019 11:06 11 Jun 2019 11:06 http://theses.gla.ac.uk/id/eprint/72607 View Item