mwang@watmath.UUCP (mwang) (03/20/84)
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_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