Numerical methods for ordinary differential equations with applications to partial differential equations

The thesis develops a number of algorithms for the numerical solution of ordinary differential equations with applications to partial differential equations.A general introduction is given; the existence of a unique solution for first order initial value problems and well known methods for analysing stability are described.A family of one-step methods is developed for first order ordinary differential equations.The methods are extrapolated and analysed for use in PECE mode and their theoretical properties, computer implementation and numerical behaviour, are discussed.La-stable methods are developed for second order parabolic partial differential equations 1n one space dimension; second and third order accuracy 1S achieved by a splitting technique 1n two space dimensions.A number of two-time level difference schemes are developed for first order hyperbolic partial differential equations and the schemes are analysed for Aa-stability and La-stability.The schemes are seen to have the advantage that the oscillations which are present with Crank-Nicolson type schemes, do not arise.CF.APTER 2 ONE -STEP METHODS FOR FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS (17)Table 2.1: Stao:lity intervals and principal error terms of the one-step multiderivative formulas.~~"ethod (Pade) Stability interval error constant -2.51,0)C4 ::I 1/ 2 4( 1,3) hE (-5.41,0)Cs = -11480(3,3) hE (-00,0) C7 -_1II ooaoo (3,2) h E (-00,0) C6 II: -117200(0,4) hE (-2.78,0)Cs • lII20(1,4) hE (-5.43,0)C6 • _1/ 3 6 0 0 (2,4) hE (-9.64,0)C7 • 1hs600 -(-19.15,0)C8 • -lii 4 11200 (3,4) h E (4,4) hE (-00,0) C9 • 1/ 2 54 0 1 6 0 0 (4,3) hE (-00,0) Ce • 111411200 (4,2) hE (-00,0) C7 • 1hs600 (4,1) h E (-00,0) C6 • 1/ 3 6 0 0 (4,0)hE (-00,0) Cs -1/120