Rational Approximation Method for Stiff Initial Value Problems

While purely numerical methods for solving ordinary differential equations (ODE), e.g., Runge–Kutta methods, are easy to implement, solvers that utilize analytical derivations of the right-hand side of the ODE, such as the Taylor series method, outperform them in many cases. Nevertheless, the Taylor series method is not well-suited for stiff problems since it is explicit and not A-stable. In our paper, we present a numerical-analytical method based on the rational approximation of the ODE solution, which is naturally A- and A(α)-stable. We describe the rational approximation method and consider issues of order, stability, and adaptive step control. Finally, through examples, we prove the superior performance of the rational approximation method when solving highly stiff problems, comparing it with the Taylor series and Runge–Kutta methods of the same accuracy order.

ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING STIFF INITIAL VALUE PROBLEMS

This paper modified an existing 3–point block method for solving stiff initial value problems. The modification leads to the derivation of another 3 – point block method which is suitable for solving stiff initial value problems. The method approximates three solutions values per step and its order is 5. Different sets of formula can be generated from it by varying a parameter ρ ϵ (-1, 1) in the formula. It has been shown that the method is both Zero stable and A–Stable. Some linear and nonlinear stiff problems are solved and the result shows that the method outperformed an existing method and competes with others in terms of accuracy

One-Point Spectrum Nordsieck Almost Collocation Methods

A family of multivalue collocation methods for the numerical solution of differen- tial problems is proposed. These methods are developed in order to be suitable for the solu- tion of stiff problems, since they are highly stable and do not suffer from order reduction, as they have uniform order of convergence in the whole integration interval. In addition, they permits to have an effcient implementation, due to the fact that the coeffcient matrix of the nonlinear sys- tem for the computation of the internal stages has a lower triangular structure with one-point spectrum. The uniform order of convergence is numerically computed in order to experimentally verify theoretical results.

High Order Relaxation Methods for Co-simulation of Finite Element and Circuit Solvers

Coupled problems result in very stiff problems whose char- acteristic parameters differ with several orders in magni- tude. For such complex problems, solving them monolithi- cally becomes prohibitive. Since nowadays there are op- timized solvers for particular problems, solving uncoupled problems becomes easy since each can be solved indepen- dently with its dedicated optimized tools. Therefore the co-simulation of the sub-problems solvers is encouraged. The design of the transmission coupling conditions between solvers plays a fundamental role. The current paper ap- plies the waveform relaxation methods for co-simulation of the finite element and circuit solvers by also investigating the contribution of higher order integration methods. The method is illustrated on a coupled finite element inductor and a boost converter and focuses on the comparison of the transmission coupling conditions based on the waveform iteration numbers between the two sub-solvers. We demon- strate that for lightly coupled systems the dynamic iterations between the sub-solvers depends much on the inter- nal integrators in individual sub-solvers whereas for tightly coupled systems it depends also to the kind of transmission coupling conditions.

First-Order Methods With Extended Stability Regions for Solving Electric Circuit Problems

Stability control of Runge-Kutta numerical schemes is studied to increase efficiency of in- tegrating stiff problems. The implementation of the algorithm to determine coefficients of stability polynomials with the use of the GMP library is presented. Shape and size of the stability region of a method can be preassigned using proposed algorithm. Sets of first-order methods with extended stability domains are built. The results of electrical circuits simulation show the increase of the efficiency of the constructed first-order methods in comparison with methods of higher order