Some new methods for stiff differential equations

Abstract

A numerical method applied to the differential equation yr=Ly can lead to a solution of the form y(x, + nh) =Azn, where z and w = Lh satisfy an equation P(w, z ) =O. In general P(w, z ) is a polynomial in both w and z, of degree M in w and N in z. Existing multistep and Rung* Kutta methods correspond to the cases M = l and N = l respectively. New methods are found by taking M 2 2 , N 22. The approach here is first to find a suitable polynomial P ( w , z ) with the desired stability properties, and then to find a process which leads to this polynomial. Third-and fourth-order -4-and L-stable processes are given of the semi-explicit and linearly implicit types.

Third-order linearly implicitl-stable methods for stiff differential equations

“…It may be noted that Process C of reference [6] is a special case of the above process with a = 0 and 7 = 1. Any process for which a = 0 gives O = O = -1.…”

“…If p is taken as 0.25, Process B of reference [6] is obtained (though written somewhat differently). Alternatively, with p=0.43 we obtain the process…”

“…The strategy used for error control is exactly as described in reference [ 6 ] . Table 1 shows the number of function evaluations (FE), Jacobian evaluations (JE), LU factorisations (LU) and steps (St) required for the various methods described above with tolerance levels (T) 0.01, 0.0001 and 0.000001.…”

“…Two methods which satisfy these criteria are given in [6]; these are particular cases of the general methods derived below.…”

GASP/S: A GASP IV version with a stiff-ODE integrator

“…As a result, 7 methods were selected that satisfy requirements of sufficient accuracy, numerical stability and computing speed. These requirements were met by semi-implicit Runge-Kutta methods (Calahan 1968;Michelsen 1977;Rosenbrock 1963;Scraton 1979), by the exponential fitted Brandon method (Babcock et al 1979), and by a little known method published by Magnus (Magnus and Schechter 1967). An implicit method called &dquo;trapezoidal rule&dquo; proved to be very robust, although somewhat slower than the other methods.…”