[ont.events] UW THESIS PRES., Mr. S.R. Trickett will speak on ``Rational Approximations to the Exponential Function for the Numerical Solution of the Heat Conduction Problem.''

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