Runge-Kutta Methods for Solving Ordinary and Delay Differential Equations

An introduction to Runge-Kutta methods for the solution of ordinary differential equations (ODEs) is introduced. The technique of using Singly Diagonally Implicit Runge-Kutta (SDIRK) method for the integration of stiff and non-stiff ODEs has been widely accepted, this is because SDIRK method is computationally efficient and stiffly stable. Consequently embedded SDIRK method of fourth-order six stage in fifth-order seven stage which has the property that the first row of the coefficient matrix is equal to zero and the last row of the coefficient matrix is equal to the vector output value is constructed. The stability region of the method when applied to linear ODE is given. Numerical results when stiff and non-stiff first order ODEs are solved using the method are tabulated and compared with the method in current use. Introduction to delay differential equations (DDEs) and the areas where they arise are given. A brief discussion on Runge-Kutta method when adapted to delay differential equation is introduced. SDIRK method which has been derived previously is used to solve delay differential equations; the delay term is approximated using divided difference interpolation. Numerical results are tabulated and compared with the existing methods. The stability aspects of SDIRK method when applied to DDEs using Lagrange interpolation are investigated and the region of stability is presented. Runge-Kutta-Nystróm (RKN) method for the solution of special second-order ordinary differential equations of the form ),(yxfy=′′ is discussed. Consequently, Singly Diagonally Implicit Runge-Kutta Nystróm (SDIRKN) method of third-order three stage embedded in fourth-order four stage with small error coefficients is constructed. The stability region of the new method is presented. The method is then used to solve both stiff and non-stiff special second order ODEs and the numerical results suggest that the new method is more efficient compared to the current methods in use. Finally, introduction to general Runge-Kutta-Nystrom (RKNG) method for the solution of second-order ordinary differential equations of the form ),,(yyxfy′=′′ is given. A new embedded Singly Diagonally Implicit Runge-Kutta-Nystróm General (SDIRKNG) method of third-order four stage embedded in fourth-order five stage is derived. Analysis on the stability aspects of the new method is given and numerical results when the method is used to solve both stiff and non-stiff second order ODEs are presented. The results indicate the superiority of the new method compared to the existing method

Asymptotically Optimal Quantum Circuits for d-level Systems

As a qubit is a two-level quantum system whose state space is spanned by |0>, |1>, so a qudit is a d-level quantum system whose state space is spanned by |0>,...,|d-1>. Quantum computation has stimulated much recent interest in algorithms factoring unitary evolutions of an n-qubit state space into component two-particle unitary evolutions. In the absence of symmetry, Shende, Markov and Bullock use Sard's theorem to prove that at least C 4^n two-qubit unitary evolutions are required, while Vartiainen, Moettoenen, and Salomaa (VMS) use the QR matrix factorization and Gray codes in an optimal order construction involving two-particle evolutions. In this work, we note that Sard's theorem demands C d^{2n} two-qudit unitary evolutions to construct a generic (symmetry-less) n-qudit evolution. However, the VMS result applied to virtual-qubits only recovers optimal order in the case that d is a power of two. We further construct a QR decomposition for d-multi-level quantum logics, proving a sharp asymptotic of Theta(d^{2n}) two-qudit gates and thus closing the complexity question for all d-level systems (d finite.) Gray codes are not required, and the optimal Theta(d^{2n}) asymptotic also applies to gate libraries where two-qudit interactions are restricted by a choice of certain architectures.Comment: 18 pages, 5 figures (very detailed.) MatLab files for factoring qudit unitary into gates in MATLAB directory of source arxiv format. v2: minor change

Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit

We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such regime the system relaxes towards a convection-diffusion equation. The first objective of the paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of the paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by an original reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis on such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, under the same conditions derived for hyperbolic relaxation. Several numerical examples including neutron transport equations confirm the theoretical analysis