The new class of A-stable hybrid multistep methods for numerical solution of stiff initial value problem

In this paper, we present a class of   hybrid multistep methods for the numerical solution of first-order initial value problems. We have used second derivative of solution (similar to second derivative multistep methods of Enright) and  an off-step point. The accuracy and stability analysis are discussed. Stability domains of our presented methods have been obtained?, ?showing that this class of efficient numerical methods are A))-stable of order up to 10. Numerical results are also given for four test problems?. Keywords: Initial value problems, Multistep methods, Off-step point, Stability aspects

Parallel iteration of the extended backward differentiation formulas

The extended backward differentiation formulas (EBDFs) and their modified form (MEBDF) were proposed by Cash in the 1980s for solving initial-value problems (IVPs) for stiff systems of ordinary differential equations (ODEs). In a recent performance evaluation of various IVP solvers, including a variable-step-variable-order implementation of the MEBDF method by Cash, it turned out that the MEBDF code often performs more efficiently than codes like RADAU5, DASSL and VODE. This motivated us to look at possible parallel implementations of the MEBDF method. Each MEBDF step essentially consists of successively solving three nonlinear systems by means of modified Newton iteration using the same Jacobian matrix. In a direct implementation of the MEBDF method on a parallel computer system, the only scope for (coarse grain) parallelism consists of a number of parallel vector updates. However, all forward-backward substitutions and all righthand side evaluations have to be done in sequence. In this paper, our starting point is the original (unmodified) EBDF method. As a consequence, two different Jacobian matrices are involved in the modified Newton method, but on a parallel computer system, the effective Jacobian-evaluation and the LU-decomposition costs are not increased. Furthermore, we consider the simultaneous solution, rather than the successive solution, of the three nonlinear systems, so that in each iteration the forward-backward substitutions and the righthand side evaluations can be done concurrently. A mutual comparison of the performance of the parallel EBDF approach and the MEBDF approach shows that we can expect a speedup factor of about 2 on 3 processors

Diagonizable extended backward differentiation formulas

We generalize the extended backward differentiation formulas (EBDFs) introduced by Cash and by Psihoyios and Cash such that the system matrix in the modified Newton process can be block-diagonalized. This enables an efficient parallel implementation. We construct methods which are L-stable up to order $p=6$ with the same computational complexity per processor as the conventional BDF methods. Numerical experiments with the order 6 method show that a speedup factor between 2 and 4 on four processors can be expected

Search of symmetric composition methods of symmetric integrators

Composition methods are useful when solving Ordinary Differential Equations (ODEs) as they increase the order of accuracy of a given basic numerical integration scheme. We will focus on sy-mmetric composition methods involving some basic second order symmetric integrator with different step sizes [17]. The introduction of symmetries into these methods simplifies the order conditions and reduces the number of unknowns. Several authors have worked in the search of the coefficients of these type of methods: the best method of order 8 has 17 stages [24], methods of order 8 and 15 stages were given in [29, 39, 40], 10-order methods of 31, 33 and 35 stages have been also found [24, 34]. In this work some techniques that we have built to obtain 10-order symmetric composition methods of symmetric integrators of s = 31 stages (16 order conditions) are explored. Given some starting coefficients that satisfy the simplest five order conditions, the process followed to obtain the coefficients that satisfy the sixteen order conditions is provided

Hybrid Linear Multistep Methods with Nested Hybrid Predictors for Solving Linear and Non-linear Initial Value Problems in Ordinary Differential Equations

In this paper, we present new class of hybrid linear multistep methods with nested hybrid predictors for the numerical integration of initial value problems in ordinary differential equations. The derivation of the method is based on interpolation and collocation procedures. The region of absolute stability of the new scheme is investigated using the boundary locus method. The method is demonstrated on some linear and non-linear problems; numerical results are tabulated and are compared with some existing methods. Key words: Interpolation, collocation, nesting, hybrid, predictors, linear multistep methods

A Comparison between High-order Temporal Integration Methods Applied to the Discontinuous Galerkin Discretized Euler Equations

Abstract In this work we present a high-order Discontinuous Galerkin (DG) space approximation coupled with two high-order temporal integration methods for the numerical solution of time-dependent compressible flows. The time integration methods analyzed are the explicit Strong-Stability-Preserving Runge-Kutta (SSPRK) and the Two Implicit Advanced Step-point (TIAS) schemes. Their accuracy and efficiency are evaluated by means of an inviscid test case for which an exact solution is available. The study is carried out for several time-steps using different polynomial order approximations and several levels of grid refinement. The effect of mesh irregularities on the accuracy is also investigated by considering randomly perturbed meshes. The analysis of the results has the twofold objective of (i) assessing the performances of the temporal schemes in the context of the high-order DG discretization and(ii) determining if high-order implicit schemes can displace widely used high-order explicit schemes