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

Solving delay differential equations by Runge-Kutta method using different types of interpolation

Introduction to delay differential equations (DDEs) and the areas where they arise are given. Analysis of specific numerical methods for solving delay differential equation is considered. A brief discussion on Runge-Kutta method when adapted to delay differential equation is introduced. Embedded Singly Diagonally Implicit Runge-Kutta (SDIRK) method of third order four-stage in fourth order five-stage which is more attractive from the practical point of view is used to solve delay differential equations. The delay term is approximated using three types of interpolation that is the divided difference interpolation, Hermite interpolation and In't Hout interpolation. Numerical results based on these three interpolations are tabulated and compared. Finally, the stability properties of SDIRK method when applied to DDEs using Lagrange interpolation and In't Hout interpolation are investigated and their regions of stability are presented

Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances

Micromagnetic Simulations of High-Speed Magnonic Devices

An emerging field of research in recent years has been magnonics, the manipulation of coherent spin excitations, spin-waves, in magnetically ordered materials. Recent advances in experimental techniques for high-frequency magnetisation dynamics and the advent of micromagnetic simulations has led to the propositions of functional magnetic devices based upon the control of spin-waves. This thesis presents work for characterisation and future development of high-speed magnonic devices derived from micromagnetic simulations, and numerical techniques for the solution of the Landau-Lifshitz equation for micromagnetic simulations in the finite-difference time-domain approach. In chapter 3, spin-waves were controlled in the propagation along a thin film magnonic waveguide via resonant scattering from a mesoscale chiral magnetic resonator, in the backwards volume, forwards volume and Damon-Eshbach geometries. The scattering interaction demonstrated non-reciprocity associated with devices acting as spin-wave diodes. Additionally, such devices demonstrated the possibility of phase-shifting. The results obtained were numerically fit and interpreted in terms of a phenomenological model of resonant chiral scattering. The origin of the chiral coupling was discussed in terms of the stray field. In chapter 4, the phenomenon of spin-wave confinement, wavelength conversion and Möbius mode formation was demonstrated in the backwards volume configuration of thin-film magnetic waveguides. The presence of magnetic field gradients or thickness gradients modified the position of the Γ-point of the dispersion relation for Backwards Volume Dipolar-Exchange Spin-Waves (BVDESW), such that back-scattering and wavelength conversion occurred from the field/thickness gradients due to the “valleys” of the spin-wave dispersion. This work highlights a basis for not only experimental observation of such phenomena, but the potential for devices based upon valleytronics, an exploitation of the valley degree-of-freedom due to the spin-wave dispersion. In chapter 5, motivated by numerical error encountered in previous work in the thesis, the validity of implicit methods formulated for the numerical solution of the Landau-Lifshitz equation for finite-difference time-domain micromagnetic simulations were demonstrated. The implicit methods were tested for single spin precession in an external field, the μMAG standard problems and additional test cases. A source of numerical instability in explicit integration methods, numerical stiffness in systems of differential equations, was demonstrated to occur in existing explicit numerical methods, applied to the Landau-Lifshitz equation, common to popular micromagnetic software. The stability of implicit methods was demonstrated to be advantageous over explicit methods in micromagnetic scenarios where numerical stiffness could occur. Additionally, it was demonstrated that the quality of the numerical results was improved compared to explicit methods when the implicit method possessed L-stability, a damping of stiff, high wave number spin waves in the simulation.Engineering and Physical Sciences Research Council (EPSRC