mwang@watmath.UUCP (mwang) (03/20/84)
delim $$ _D_E_P_A_R_T_M_E_N_T _O_F _C_O_M_P_U_T_E_R _S_C_I_E_N_C_E _U_N_I_V_E_R_S_I_T_Y _O_F _W_A_T_E_R_L_O_O _S_E_M_I_N_A_R _A_C_T_I_V_I_T_I_E_S _T_H_E_S_I_S _P_R_E_S_E_N_T_A_T_I_O_N - Thursday, March 22, 1984. Mr. S.R. Trickett, a graduate student of this depart- ment, will speak on ``Rational Approximations to the Exponential Function for the Numerical Solution of the Heat Conduction Problem.'' TIME: 3:30 PM ROOM: MC 6091A ABSTRACT A common approach for the numerical solution of the heat conduction problem is to reduce it to a set of or- dinary differential equations through discretization of the space variables. Unfortunately the resulting sys- tem displays certain properties, particularly sparse- ness and stiffness, which make it unsuited to solution by many standard numerical ODE solvers. The methods examined here are an alternate formulation of the semi-implicit Runga-Kutta schemes, whereby the problem is further reduced to that of estimating an exponential function involving matrix arguments. Implementation considerations lead us to consider rational approxima- tions of the form $ e sup -z ~approx~(c sub 0 ~+~c sub 1 z ~+~...~+~c sub m z sup m )~/~(1~+~bz) sup n ,~m~<=~n~$, where the value of $ b $ is chosen first, and the numerator coefficients are then calculated to give max- imum order at zero. The primary aim of this thesis is to determine what values of the parameter $ b $ most benefit the solution of the heat conduction problem. We find that by making the method exact for some critical eigenvalue of the complementary problem, performance during the transient phase can be greatly enhanced. The ability of the approximations to satisfy such a criterion is esta- blished by two existence theorems. Results concerning $ roman {A} sub 0 $-stability and the attenuation of high frequency components are also given. Finally, a physical problem involving heat conduction in a thermal print head is used to more fully demonstrate the behaviour of these methods. This is a formal presentation of a thesis and will be followed by an examination. March 20, 1984