Numerical Solution for Kawahara Equation by Using Spectral Methods

AbstractSome nonlinear wave equations are more difficult to investigate mathematically, as no general analytical method for their solutions exists. The Exponential Time Differencing (ETD) technique requires minimum stages to obtain the requiredaccurateness, which suggests an efficient technique relatingto computational duration thatensures remarkable stability characteristicsupon resolving nonlinear wave equations. This article solves the diagonal example of Kawahara equation via the ETD Runge-Kutta 4 technique. Implementation of this technique is proposed by short Matlab programs

An Algorithmic Framework for Efficient Large-Scale Circuit Simulation Using Exponential Integrators

We propose an efficient algorithmic framework for time domain circuit simulation using exponential integrator. This work addresses several critical issues exposed by previous matrix exponential based circuit simulation research, and makes it capable of simulating stiff nonlinear circuit system at a large scale. In this framework, the system's nonlinearity is treated with exponential Rosenbrock-Euler formulation. The matrix exponential and vector product is computed using invert Krylov subspace method. Our proposed method has several distinguished advantages over conventional formulations (e.g., the well-known backward Euler with Newton-Raphson method). The matrix factorization is performed only for the conductance/resistance matrix G, without being performed for the combinations of the capacitance/inductance matrix C and matrix G, which are used in traditional implicit formulations. Furthermore, due to the explicit nature of our formulation, we do not need to repeat LU decompositions when adjusting the length of time steps for error controls. Our algorithm is better suited to solving tightly coupled post-layout circuits in the pursuit for full-chip simulation. Our experimental results validate the advantages of our framework.Comment: 6 pages; ACM/IEEE DAC 201

On the numerical solution of linear stiff IVPs by modified homotopy perturbation method

In this paper, we introduce a method to solve linear sti® IVPs. The sug-gested method, which we call modi¯ed homotopy perturbation method, can be considered as an extension of the homotopy perturbation method (HPM) which is very efficient in solving a varety of di®erential and algebraic equations. In this work, a class of linear stiff initial value problems (IVPs) are solved by the classical homotopy per-turbation method (HPM), modified homotopy perturbation method and an explicit Runge-Kutta-type method (RK). Numerical comparisons demonstrate the limitations of HPM and promising capability of the MHPM for solving stiff IVPs. The results prove that the modified HPM is a powerful tool for the solution of linear stiff IVPs

Solving linear and non-linear stiff system of ordinary differential equations by multistage adomian decomposition method

In this paper, linear and non-linear stiff systems of ordinary differential equations are solved by the classical Adomian decomposition method (ADM) and the multistage Adomian decomposition method (MADM). The MADM is a technique adapted from the standard Adomian decomposition method (ADM) where standard ADM is converted into a hybrid numeric-analytic method called the multistage ADM (MADM). The MADM is tested for several examples. Comparisons with an explicit Runge-Kutta-type method (RK) and the classical ADM demonstrate the limitations of ADM and promising capability of the MADM for solving stiff initial value problems (IVPs)

Експоненціальна дискретизація задачі Коші для звичайних диференціальних рівнянь

Використовуючи однокроковоу рекурентну схему з експоненціальними ваговими функціями побудовано чисельну апроксимацію для задачі Коші. Показано здатність такої схеми точно відтворювати вузлові значення шуканого розв’язку. Проведено аналіз відносно стійкості та збіжності, а також порівняно властивості запропонованої схеми з методом Кранка-Ніколсона. Подано числові результати розрахунку для сингулярно збуреної задачі Коші із застосуванням експоненціальної схеми.Numerical approximation of the Cauchy problem has been suggested on the basis of one-step recurrent integration scheme with exponential weighting functions. Its ability to reproduce exact nodal values of the solution has been demonstrated. The analysis of consistency and convergence of the scheme are considered. Properties of proposed scheme has been compared with one based on Crank-Nicolson method. The numerical solution of singularly perturbed problem found by the scheme has been presented.Используя одношаговою рекурентную схему с экспоненциальніми весовыми функциями построено числовую аппроксимацию для задачи Коши. Показано возможность такой схемы точно воспроизводить узловые значения искомого решения. Проведен анализ относительно стойкости и сходимости, а также сравнение свойств предложенной схемы с методом Кранка-Николсона. Представлены числовые результаты расчета для сингулярно возмущенной задачи Коши с применением експоненциальной схемы

A fast time-domain EM-TCAD coupled simulation framework via matrix exponential

We present a fast time-domain multiphysics simulation framework that combines full-wave electromagnetism (EM) and carrier transport in semiconductor devices (TCAD). The proposed framework features a division of linear and nonlinear components in the EM-TCAD coupled system. The former is extracted and handled independently with high efficiency by a matrix exponential approach assisted with Krylov subspace method. The latter is treated by ordinary Newton's method yet with a much sparser Jacobian matrix that leads to substantial speedup in solving the linear system of equations. More convenient error management and adaptive control are also available through the linear and nonlinear decoupling. © 2012 ACM.published_or_final_versio

Cubic autocatalysis in a reaction-diffusion annulus: semi-analytical solutions

Semi-analytical solutions for cubic autocatalytic reactions are considered in a circularly symmetric reaction-diffusion annulus. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for this novel geometry. Singularity theory is used to determine the regions of parameter space in which the different types of steady-state diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found using a degenerate Hopf bifurcation analysis. A novel feature of this geometry is the effect, of varying the width of the annulus, on the static and dynamic multiplicity. The results show that for a thicker annulus, Hopf bifurcations and multiple steady-state solutions occur in a larger portion of parameter space. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations

Numerical Methods for Two-Dimensional Stem Cell Tissue Growth.

Growth of developing and regenerative biological tissues of different cell types is usually driven by stem cells and their local environment. Here, we present a computational framework for continuum tissue growth models consisting of stem cells, cell lineages, and diffusive molecules that regulate proliferation and differentiation through feedback. To deal with the moving boundaries of the models in both open geometries and closed geometries (through polar coordinates) in two dimensions, we transform the dynamic domains and governing equations to fixed domains, followed by solving for the transformation functions to track the interface explicitly. Clustering grid points in local regions for better efficiency and accuracy can be achieved by appropriate choices of the transformation. The equations resulting from the incompressibility of the tissue is approximated by high-order finite difference schemes and is solved using the multigrid algorithms. The numerical tests demonstrate an overall spatiotemporal second-order accuracy of the methods and their capability in capturing large deformations of the tissue boundaries. The methods are applied to two biological systems: stratified epithelia for studying the effects of two different types of stem cell niches and the scaling of a morphogen gradient with the size of the Drosophila imaginal wing disc during growth. Direct simulations of both systems suggest that that the computational framework is robust and accurate, and it can incorporate various biological processes critical to stem cell dynamics and tissue growth