Parallel Diagonally Implicit Runge-Kutta Methods For Solving Ordinary Differential Equations

This thesis focuses on the derivations of diagonally implicit Runge-Kutta (DIRK) methods with the capability to be implemented by parallel executions. A few new methods are proposed by having sparsity patterns which enable the parallelization of methods. In the first part of the thesis, a fifth order DIRK suitable for two processors parallel executions and DIRK methods of fourth and fifth orders suitable for three processors are proposed. The executions of these methods are done by using fixed stepsizes on a set of nonstiff problems. The regions of stability are presented and numerical results of the methods are compared to the existing methods. Parallel computations show significant time reduction when solving large systems of nonstiff ordinary differential equations (ODEs). The subsequent part of the thesis discusses on embedded DIRK methods suitable for two processors implementations. Two 4(3) and also two 5(4) embedded DIRK methods with adequate stability regions to solve stiff ODEs are proposed. Numerical experiments on stiff test problems are done based on variable stepsize strategy. An existing code for solving stiff ODEs suitable for embedded DIRK with equal diagonal elements is modified to accommodate the new methods with alternate diagonal elements. Comparisons on numerical results to existing methods show a competitive efficiency when solving small systems of stiff ODEs. A parallel code is developed with the same capability of the modified sequential code to handle stiff ODEs, linear and nonlinear problems. All algorithms are written in C language and the parallel code is implemented on Sun Fire V1280 distributed memory system. Three large scales of stiff ODEs are used to measure the parallel performances of the new embedded methods. Results show that speedups increased as the dimensions of the problems gets larger which is a significant contribution in reducing the cost of computations

Parallel-iterated Runge-Kutta methods for stiff ordinary differential equations

AbstractFor the numerical integration of a stiff ordinary differential equation, fully implicit Runge-Kutta methods offer nice properties, like a high classical order and high stage order as well as an excellent stability behaviour. However, such methods need the solution of a set of highly coupled equations for the stage values and this is a considerable computational task. This paper discusses an iteration scheme to tackle this problem. By means of a suitable choice of the iteration parameters, the implicit relations for the stage values, as they occur in each iteration, can be uncoupled so that they can be solved in parallel. The resulting scheme can be cast into the class of Diagonally Implicit Runge-Kutta (DIRK) methods and, similar to these methods, requires only one LU factorization per step (per processor). The stability as well as the computational efficiency of the process strongly depends on the particular choice of the iteration parameters and on the number of iterations performed. We discuss several choices to obtain good stability and fast convergence. Based on these approaches, we wrote two codes possessing local error control and stepsize variation. We have implemented both codes on an ALLIANT FX/4 machine (four parallel vector processors and shared memory) and measured their speedup factors for a number of test problems. Furthermore, the performance of these codes is compared with the performance of the best stiff ODE codes for sequential computers, like SIMPLE, LSODE and RADAU5

Computer solution of non-linear integration formula for solving initial value problems

This thesis is concerned with the numerical solutions of initial value problems with ordinary differential equations and covers single step integration methods. focus is to study the numerical the various aspects of Specifically, its main methods of non-linear integration formula with a variety of means based on the Contraharmonic mean (C˳M) (Evans and Yaakub [1995]), the Centroidal mean (C˳M) (Yaakub and Evans [1995]) and the Root-Mean-Square (RMS) (Yaakub and Evans [1993]) for solving initial value problems. the applications of the second It includes a study of order C˳M method for parallel implementation of extrapolation methods for ordinary differential equations with the ExDaTa schedule by Bahoshy [1992]. Another important topic presented in this thesis is that a fifth order five-stage explicit Runge Kutta method or weighted Runge Kutta formula [Evans and Yaakub [1996]) exists which is contrary to Butcher [1987] and the theorem in Lambert ([1991] ,pp 181). The thesis is organized as follows. An introduction to initial value problems in ordinary differential equations and parallel computers and software in Chapter 1, the basic preliminaries and fundamental concepts in mathematics, an algebraic manipulation package, e.g., Mathematica and basic parallel processing techniques are discussed in Chapter 2. Following in Chapter 3 is a survey of single step methods to solve ordinary differential equations. In this chapter, several single step methods including the Taylor series method, Runge Kutta method and a linear multistep method for non-stiff and stiff problems are also considered. Chapter 4 gives a new Runge Kutta formula for solving initial value problems using the Contraharmonic mean (C˳M), the Centroidal mean (C˳M) and the Root-MeanSquare (RMS). An error and stability analysis for these variety of means and numerical examples are also presented. Chapter 5 discusses the parallel implementation on the Sequent 8000 parallel computer of the Runge-Kutta contraharmonic mean (C˳M) method with extrapolation procedures using explicit assignment scheduling Kutta RK(4, 4) method (EXDATA) strategies. A is introduced and the data task new Rungetheory and analysis of its properties are investigated and compared with the more popular RKF(4,5) method, are given in Chapter 6. Chapter 7 presents a new integration method with error control for the solution of a special class of second order ODEs. In Chapter 8, a new weighted Runge-Kutta fifth order method with 5 stages is introduced. By comparison with the currently recommended RK4 ( 5) Merson and RK5(6) Nystrom methods, the new method gives improved results. Chapter 9 proposes a new fifth order Runge-Kutta type method for solving oscillatory problems by the use of trigonometric polynomial interpolation which extends the earlier work of Gautschi [1961]. An analysis of the convergence and stability of the new method is given with comparison with the standard Runge-Kutta methods. Finally, Chapter 10 summarises and presents conclusions on the topics discussed throughout the thesis

A parallel nearly implicit time-stepping scheme

Across-the-space parallelism still remains the most mature, convenient and natural way to parallelize large scale problems. One of the major problems here is that implicit time stepping is often difficult to parallelize due to the structure of the system. Approximate implicit schemes have been suggested to circumvent the problem. These schemes have attractive stability properties and they are also very well parallelizable.\ud The purpose of this article is to give an overall assessment of the parallelism of the method

Multi-Adaptive Time-Integration

Time integration of ODEs or time-dependent PDEs with required resolution of the fastest time scales of the system, can be very costly if the system exhibits multiple time scales of different magnitudes. If the different time scales are localised to different components, corresponding to localisation in space for a PDE, efficient time integration thus requires that we use different time steps for different components. We present an overview of the multi-adaptive Galerkin methods mcG(q) and mdG(q) recently introduced in a series of papers by the author. In these methods, the time step sequence is selected individually and adaptively for each component, based on an a posteriori error estimate of the global error. The multi-adaptive methods require the solution of large systems of nonlinear algebraic equations which are solved using explicit-type iterative solvers (fixed point iteration). If the system is stiff, these iterations may fail to converge, corresponding to the well-known fact that standard explicit methods are inefficient for stiff systems. To resolve this problem, we present an adaptive strategy for explicit time integration of stiff ODEs, in which the explicit method is adaptively stabilised by a small number of small, stabilising time steps

Extrapolation-Based Implicit-Explicit Peer Methods with Optimised Stability Regions

In this paper we investigate a new class of implicit-explicit (IMEX) two-step methods of Peer type for systems of ordinary differential equations with both non-stiff and stiff parts included in the source term. An extrapolation approach based on already computed stage values is applied to construct IMEX methods with favourable stability properties. Optimised IMEX-Peer methods of order p = 2, 3, 4, are given as result of a search algorithm carefully designed to balance the size of the stability regions and the extrapolation errors. Numerical experiments and a comparison to other implicit-explicit methods are included.Comment: 21 pages, 6 figure

Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part

In this paper, we extend the implicit-explicit (IMEX) methods of Peer type recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203--215, 2017] to a broader class of two-step methods that allow the construction of super-convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes combine the necessary stability of implicit and low computational costs of explicit methods to efficiently solve systems of ordinary differential equations with both stiff and non-stiff parts included in the source term. To construct super-convergent IMEX-Peer methods with favourable stability properties, we derive necessary and sufficient conditions on the coefficient matrices and apply an extrapolation approach based on already computed stage values. Optimised super-convergent IMEX-Peer methods of order s+1 for s=2,3,4 stages are given as result of a search algorithm carefully designed to balance the size of the stability regions and the extrapolation errors. Numerical experiments and a comparison to other IMEX-Peer methods are included.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with arXiv:1610.0051

Task-based adaptive multiresolution for time-space multi-scale reaction-diffusion systems on multi-core architectures

A new solver featuring time-space adaptation and error control has been recently introduced to tackle the numerical solution of stiff reaction-diffusion systems. Based on operator splitting, finite volume adaptive multiresolution and high order time integrators with specific stability properties for each operator, this strategy yields high computational efficiency for large multidimensional computations on standard architectures such as powerful workstations. However, the data structure of the original implementation, based on trees of pointers, provides limited opportunities for efficiency enhancements, while posing serious challenges in terms of parallel programming and load balancing. The present contribution proposes a new implementation of the whole set of numerical methods including Radau5 and ROCK4, relying on a fully different data structure together with the use of a specific library, TBB, for shared-memory, task-based parallelism with work-stealing. The performance of our implementation is assessed in a series of test-cases of increasing difficulty in two and three dimensions on multi-core and many-core architectures, demonstrating high scalability