Local Linearization—Runge–Kutta methods: A class of A-stable explicit integrators for dynamical systems

Abstract

a b s t r a c tA new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of the original equation in two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary ODE. The first one is solved by a Local Linearization scheme in such a way that A-stability is ensured, while the second one can be approximated by any extant scheme, preferably a high order explicit Runge-Kutta scheme. Results on the convergence and dynamical properties of this new class of schemes are given, as well as some hints for their efficient numerical implementation. An specific scheme of this new class is derived in detail, and its performance is compared with some Matlab codes in the integration of a variety of ODEs representing different types of dynamics.

Numerical solution of stochastic quantum master equations using stochastic interacting wave functions

“…If A is a large and sparse matrix and v is a vector, then we can get exp (A) v by Krylov subspace iterative methods (see, e.g., [22,28,56]). In case the dimension of v is less than a few thousand, we use the standard scaling and squaring method with Padé approximants for computing the exponential of A on a current computer (see, e.g., [20,26,27,39]). In the latter method, m ∈ Z + is chosen such that A/2 m is of order of magnitude 1 and exp (A/2 m ) is approximated by the rational function D pq (A/2 m ) −1 N pq (A/2 m ), where p, q ∈ N and for any matrix X we define…”

“…Fix p, q ∈ N and m ∈ Z + . For any matrix X we set P pq (X) = D pq (X) −1 N pq (X), where D pq and N pq are as in (20). Then…”

“…Suppose that D pq and N pq are given by (20) and that P pq , N pq , D pq are as in Theorem 3.1. Then…”

A class of high-order Runge–Kutta–Chebyshev stability polynomials

“…This property may be exploited in solving numerical problems for semi-linear stiff systems of equations through operator splitting methods [37,38,39,40]. We note that in the literature, the linear and nonlinear terms of reaction-diffusion models have been decoupled under a variety of numerical integration techniques including: splitting methods [41, and previous references], Implicit-Explicit Runge-Kutta-Chebyshev (IMEX RKC) methods [42,43], PIROCK [44], and Local Linearization Runge-Kutta (LLRK) methods [45,46].…”

A New Method to Solve Numeric Solution of Nonlinear Dynamic System

“…After years of accumulation and development, there are too many methods to solve the numerical solution of nonlinear dynamic systems; the main methods are as follows: perturbation method [3], averaging method [4], Runge-Kutta method [5], Euler method [6], gradient method [7] and so on. Regretfully, these methods have certain advantages in solving certain system but obtain unappealing outcomes when solving problems of general nonlinear dynamic systems, like the lower precision, the complicity and large calculation quantity, Runge phenomenon, and so forth.…”