Explicit Time Integration Scheme Using Krylov Subspace Method for Reactor Kinetics Equation

Abstract

The spatial discretization form of the space-dependent reactor kinetics equation is a first-order simultaneous ordinary differential equation in time. Conventional numerical methods of the space-dependent kinetics equation, i.e., the generalized Runge-Kutta method, the implicit method (backward Euler method), and the Theta method, are based on the time difference approximation. However, the present study adopts the analytical solution of the space-dependent kinetics equation expressed by the matrix exponential and no time difference approximation is used. In this context, our present approach is classified as an explicit method in which no iteration calculation on space and energy is necessary. The Krylov subspace method is used to evaluate the matrix exponential observed in the solution of the spatially discretized space-dependent kinetics equation. The Krylov subspace method is implemented into a space-dependent kinetics solver. In order to examine the effectiveness of the Krylov subspace method, the TWIGL benchmark problem is analyzed as a verification calculation. The calculation results show the effectiveness of the present method especially in the step reactivity perturbation.

Numerical solutions of the time‐dependent Schrödinger equation with position‐dependent effective mass

“…The basic idea of the Krylov subspace approach is to project the exponential of a large matrix/operator onto a relatively small-sized Krylov subspace where calculating the exponential is significantly less computationally expensive [63]. The Krylov subspace method-based exponential integration has been applied successfully for solving many different problems [13,22,32,35,69], especially in differential equations, such as Maxwell's equations in time [13,15,56], large system of differential equations [35], multifrequency optical response [14], reactor kinetics equation [4], fast pricing of options equations [71], fluid dynamics equations [64], shallow water equations [30], Dirac equation [9], incompressible Navier-Stokes equations [21], etc. We shall apply it for solving the subproblem of Equation ( 1) that is related to the kinetic operator as well.…”

Matrix exponential based electromagnetic transients simulation algorithm with Krylov subspace approximation and accurate dense output

“…This analytical solution methodology avoids the local truncation error that is encountered in traditional integration methods, and can be generalized for nonlinear systems by utilizing Jacobian information of nonlinearity [3] [4]. Matrix exponential based integration methods have found application in several engineering fields, including chemical engineering [5], computational physics [6], and large scale integration circuit analysis [7]. This paper proposes an adapted matrix exponential method that is suitable for power system electromagnetic transients simulation, especially in the following three aspects: (1) high numerical accuracy and A-stability, which enables handling typical stiff power system simulation models; (2) subspace approximation method of exponential operator, which increases algorithm scalability for solving large scale problem; (3) accurate dense output functionality, which makes possible the adoption of larger simulation step size and revealing higher frequency dynamic details.…”