A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

              المشتقة الثانية طريقة رنك-كوتا الفعالة الجديدة من الرتبة الخامسة  (TDRK) قد تم تطويرها من أجل الحل العددي للمعادلات التفاضلية الاعتيادية من الرتبة الأولى (ODEs). تم اشتقاق الطريقة الجديدة باستخدام خاصية الأول  نفس الأخير  (FSAL) . قمنا بتحليل استقرار الطريقة. تم عرض النتائج العددية لتوضيح كفاءة الطريقة الجديدة بالمقارنة مع بعض طرق رنك-كوتا (RK) المعروفة.A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods

Splitting methods for autonomous and non-autonomous perturbed equations

[EN] This thesis addresses the treatment of perturbed problems with splitting methods. After motivating these problems in Chapter 1, we give a thorough introduction in Chapter 2, which includes the objectives, several basic techniques and already existing methods. In Chapter 3, we consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative coefficients). We propose to consider a class of methods that allows us to evaluate all time dependent operators at real values of the time, leading to schemes which are stable and simple to implement. If the system can be considered as the perturbation of an exactly solvable problem and the flow of the dominant part is advanced using real coefficients, it is possible to build highly efficient methods for these problems. We show the performance of this class of methods for several numerical examples and present some new improved schemes. In Chapter 4, we propose splitting methods for the computation of the exponential of perturbed matrices which can be written as the sum A = D+epsilon*B of a sparse and efficiently exponentiable matrix D with sparse exponential exp(D) and a dense matrix epsilon*B which is of small norm in comparison with D. The predominant algorithm is based on scaling the large matrix A by a small number 2^(-s) , which is then exponentiated by efficient Padé or Taylor methods and finally squared in order to obtain an approximation for the full exponential. In this setting, the main portion of the computational cost arises from dense-matrix multiplications and we present a modified squaring which takes advantage of the smallness of the perturbation matrix B in order to reduce the number of squarings necessary. Theoretical results on local error and error propagation for splitting methods are complemented with numerical experiments and show a clear improvement over existing methods when medium precision is sought. In Chapter 5, we consider the numerical integration of the perturbed Hill's equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, the Hill's equations originate from a Hamiltonian function and the fundamental matrix solution is a symplectic matrix. This is a very important property to be preserved by the numerical integrators. In this chapter we present new sixth-and eighth-order symplectic exponential integrators that are tailored to the Hill's equation. The methods are based on an efficient symplectic approximation to the exponential of high dimensional coupled autonomous harmonic oscillators and yield accurate results for oscillatory problems at a low computational cost. Several numerical examples illustrate the performance of the new methods. Conclusions and pointers to further research are detailed in Chapter 6.[ES] Esta tesis aborda el tratamiento de problemas perturbados con métodos de escisión (splitting). Tras motivar el origen de este tipo de problemas en el capítulo 1, introducimos los objetivos, varias técnicas básicas y métodos existentes en capítulo 2. En el capítulo 3 consideramos la integración numérica de ecuaciones no autónomas separables y parabólicas usando métodos de splitting de orden mayor que dos usando coeficientes complejos (métodos con coeficientes reales de orden mayor de dos necesariamente tienen coeficientes negativos). Proponemos una clase de métodos que permite evaluar todos los operadores con dependencia temporal en valores reales del tiempo lo cual genera esquemas estables y fáciles de implementar. Si el sistema se puede considerar como una perturbación de un problema resoluble de forma exacta y si el flujo de la parte dominante se avanza usando coeficientes reales, es posible construir métodos altamente eficientes para este tipo de problemas. Demostramos la eficiencia de estos métodos en varios ejemplos numéricos. En el capítulo 4 proponemos métodos de splitting para el cálculo de la exponencial de matrices perturbadas que se pueden escribir como suma A = D + epsilon*B de una matriz dispersa y eficientemente exponenciable con exponencial dispersa exp(D) y una matriz densa epsilon*B de noma pequeña. El algoritmo predominante se basa en escalar la matriz grande con un número pequeño 2^(-s) para poder exponenciar el resultado con métodos eficientes de Padé o Taylor y finalmente obtener la aproximación a la exponencial elevando al cuadrado repetidamente. En este contexto, el coste computacional proviene de las multiplicaciones de matrices densas y presentamos una cuadratura modificada aprovechando la estructura perturbada para reducir el número de productos. Resultados teóricos sobre errores locales y propagación de error para métodos de splitting son complementados con experimentos numéricos y muestran una clara mejora sobre métodos existentes a precisión media. En el capítulo 5, consideramos la integración numérica de la ecuación de Hill perturbada. Resonancias paramétricas pueden aparecer y esta propiedad es de gran interés en muchas aplicaciones físicas. Habitualmente, las ecuaciones de Hill provienen de una función hamiltoniana y la solución fundamental es una matriz simpléctica, una propiedad muy importante que preservar con los integradores numéricos. Presentamos nuevos integradores simplécticos exponenciales de orden seis y ocho tallados a la ecuación de Hills. Estos métodos se basan en una aproximación simpléctica eficiente a la exponencial de osciladores armónicos acoplados de dimensión alta y dan lugar a resultados precisos para problemas oscilatorios a un coste computacional bajo y varios ejemplos numéricos ilustran su rendimiento. Conclusiones e indicadores para futuros estudios se detallan en el capítulo 6.[CA] La present tesi està enfocada al tractament de problemes perturbats utilitzant, entre altres, mètodes d'escisió (splitting). Comencem motivant l'oritge d'aquest tipus de problems al capítol 1, i a continuació introduïm el objectius, diferents tècniques bàsiques i alguns mètodes existents al capítol 2. Al capítol 3, consideram la integració numèrica d'equacions no autònomes separables i parabòliques utilitzant mètodes d'splitting d'ordre major que dos utilitzant coeficients complexos (mètodes amb coeficients reials d'ordre major que dos necesariament tenen coeficients negatius). Proposem una clase de mètodes que permeten evaluar tots els operadors amb dependència temporal explícita amb valors reials del temps. Esta forma de procedir genera esquemes estables i fàcils d'implementar. Si el sistema es pot considerar com una perturbació d'un problema exactament resoluble, i la part dominant s'avança utilitzant coeficients reials, es posible construir mètodes altament eficients per aquest tipus de problemes Demostrem la eficiència d'estos mètodes per a diferents exemples numèrics. Al capítol 4, proposem mètodes d'splitting per al càcul de la exponencial de matrius pertorbades que es poden escriure com suma A = D + epsilon*B (una matriu que es pot exponenciar fàcilment i eficientemente, com es el cas d'algunes matrius disperses exp(D), i una matriu densa epsilon*B de norma menuda). L'algorisme predominant es basa en escalar la matriu gran amb un nombre menut 2^(-s) per a poder exponenciar el resultat amb mètodes eficients de Padé o Taylor i finalment obtindre la aproximació a la exponencial elevant al quadrat repetidament. En este context, el cost computacional prové de les multiplicacions de matrius denses i presentem una quadratura modificada aprofitant la estructura de matriu pertorbada per reduir el nombre de productes. Resultats teòrics sobre errors locals i propagació d'error per a mètodes d'splitting son analitzats i corroborats amb experiments numèrics, mostrant una clara millora respecte a mètodes existens quan es busca una precisió moderada. Al capítol 5, considerem la integració numèrica de l'ecuació de Hill pertorbada. En este tipus d'equacions poden apareixer resonàncies paramètriques i esta propietat es de gran interés en moltes aplicacions físiques. Habitualment, les equacions de Hill provenen d'una función hamiltoniana i la solució fonamental es una matriu simplèctica, siguent esta una propietat molt important a preservar pels integradors numèrics. Presentams nous integradors simplèctics exponencials d'orden sis i huit construits especialmente per resoldre l'ecuació de Hill. Estos mètodes es basen en una aproxmiació simplèctica eficient a la exponencial d'osciladors harmònics acoplats de dimensió alta i donen lloc a resultats precisos per a problemas oscilatoris a un cost computacional baix. La eficiencia dels mètodes s'il.lustra en diferents exemples numèrics. Conclusions i indicadors per a futurs estudis es detallen al capítol 6.Seydaoglu, M. (2016). Splitting methods for autonomous and non-autonomous perturbed equations [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/71358TESI

Geometric Integrators for Schrödinger Equations

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control.Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716TESISPremios Extraordinarios de tesis doctorale

Collocation methods for a class of second order initial value problems with oscillatory solutions

We derive and analyse two families of multistep collocation methods for periodic initial-value problems of the form y" = f(x, y); y((^x)o) = yo, y(^1)(xo) = zo involving ordinary differential equations of second order in which the first derivative does not appear explicitly. A survey of recent results and proposed numerical methods is given in chapter 2. Chapter 3 is devoted to the analysis of a family of implicit Chebyshev methods proposed by Panovsky k Richardson. We show that for each non-negative integer r, there are two methods of order 2r from this family which possess non-vanishing intervals of periodicity. The equivalence of these methods with one-step collocation methods is also established, and these methods are shown to be neither P-stable nor symplectic. In chapters 4 and 5, two families of multistep collocation methods are derived, and their order and stability properties are investigated. A detailed analysis of the two-step symmetric methods from each class is also given. The multistep Runge-Kutta-Nystrom methods of chapter 4 are found to be difficult to analyse, and the specific examples considered are found to perform poorly in the areas of both accuracy and stability. By contrast, the two-step symmetric hybrid methods of chapter 5 are shown to have excellent stability properties, in particular we show that all two-step 27V-point methods of this type possess non-vanishing intervals of periodicity, and we give conditions under which these methods are almost P-stable. P-stable and efficient methods from this family are obtained and demonstrated in numerical experiments. A simple, cheap and effective error estimator for these methods is also given

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