Explicit exponential method for the integration of stiff ordinary differential equations

Abstract

A new procedure for the integration of both stiff and nonstiff ordinary differential equations is presented. This new approach, which is applicable to general nonlinear systems, is fully explicit, requires little computer memory, and is very easy to program. It does not require computation of a Jacobian matrix or the solution of algebraic equations. The new algorithm has much better stability properties than the classical explicit methods and is generally much more efficient than these methods for stiff problems. The new approach automatically partitions the dependent variables at every step into stiff and nonstiff groups, and subsequently integrates each accordingly, using a nonstiff method for the nonstiff variables and a new explicit exponential method for the stiff variables. The algorithm blends these approximations appropriately. Because of the automatic partitioning, the new algorithm is entirely suitable for both stiff and nonstiff problems. Nomenclature= derivative value for /th variable at start of RK2 step = derivative value for /th variable at end of RK2 step = arbitrary function in Eq. (2) = derivative vector of y,y ' = f(t,y),= integration step size = initial integration step size = exponential or "stiffness" parameter for /th variable -number of first-order ordinary differential equations in system = independent variable = temporary local exponentially fitted solution for /th variable, used to estimate local error for the exponential method, Eq. (7) = solution vector for the dependent variables, Oi, • • • , >W) r = vector of initial values for the dependent variables at initial point t 0 , b>i(f 0 ), . . . , y N (t 0 )] T = temporary local Euler solution vector, used to estimate local error for RK2 = positive constant in prototype equation, y ' = -\y

Explicit adaptive Runge-Kutta methods

“…The step of integration is performed according to the formula where is the vector of the parameters being adjusted. Taking (5) into account, this formula can be writ ten as (6) where . Hereinafter, we will use formula (6), as it has a fewer number of terms and is less sensitive to calculation errors.…”

“…(7), from which we obtain (10) Formulas (9) and (10) set a method which is the initial method in the two step Fowler-Warten method [1]. Methods of this type were considered also in [5,6]; they are called exponential.…”

“…Such estimates are easy to obtain by the results of stages of the explicit method, which requires almost no additional calculations. Such an approach was implemented in explicit nonlinear methods [1][2][3][4][5][6][7]. We will note among them the method of Fowler-Warten [1] who were the first to identify the stages in estimates of eigenvalues and the subsequent use of these estimates for stabilization of the calculation scheme.…”