A linearly semi-implicit compact scheme for the Burgers–Huxley equation

In this paper, a linearly semi-implicit compact scheme is developed for the Burgers–Huxley equation. The equation is decomposed into two subproblems, i.e. a Burgers equation and a nonlinear ODE, by the operator splitting technique. The Burgers equation is solved by a linearly self-starting compact scheme which is fourth-order accurate in space and second-order accurate in time. The nonlinear ODE is discretized by a third-order semi-implicit Runge–Kutta method, which possesses good numerical stability with low computational cost. The numerical experiments show that the scheme provides the expected convergence order. Finally, several experiments are conducted to simulate the solutions of the Burgers–Huxley equation to validate our numerical method.     

基于HH模型的心肌细胞电模型

提出了一种心肌细胞电生理模型的建模仿真方法。通过借助Hodgkin-Huxley模型对单个心肌细胞建立细胞膜的等效电路模型,利用四阶Rouge-Kutta算法,研究并分析了心肌细胞膜内外离子电流及电位差变化。然后用C语言完成了对细胞膜等效电路模型的编程,利用Matlab软件平台进行了计算机仿真,得到了心肌细胞电生理学模型在不同刺激下的仿真实验结果。     

赫胥黎的科学普及思想

赫胥黎是科学普及发展历程中的著名人物,他通过大量的科普讲座和科普读物,让公众了解并热爱科学,促进了科学普及事业的发展。本文从主体、对象、目的、途径等角度,对赫胥黎科学普及思想进行了探讨。     

Numerical solutions of generalized Burgers–Fisher and generalized Burgers–Huxley equations using collocation of cubic B-splines

In this work, we propose a numerical scheme to obtain approximate solutions of generalized Burgers–Fisher and Burgers–Huxley equations. The scheme is based on collocation of modified cubic B-spline functions and is applicable for a class of similar diffusion–convection–reaction equations. We use modified cubic B-spline functions for space variable and for its derivatives to obtain a system of first-order ordinary differential equations in time. We solve this system by using SSP-RK54 scheme. The stability of the method has been discussed and it is shown that the method is unconditionally stable. The approximate solutions have been computed without using any transformation or linearization. The proposed scheme needs less storage space and execution time. The test problems considered by the different researchers have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in the literature. The scheme is simple as well as computationally efficient. The scheme provides approximate solution not only at the grid points but also at any point in the solution range.     

A multi‐GPU algorithm for large‐scale neuronal networks

Large‐scale simulations of parts of the brain using detailed neuronal models to improve our understanding of brain functions are becoming a reality with the usage of supercomputers and large clusters. However, the high acquisition and maintenance cost of these computers, including the physical space, air conditioning, and electrical power, limits the number of simulations of this kind that scientists can perform. Modern commodity graphical cards, based on the CUDA platform, contain graphical processing units (GPUs) composed of hundreds of processors that can simultaneously execute thousands of threads and thus constitute a low‐cost solution for many high‐performance computing applications. In this work, we present a CUDA algorithm that enables the execution, on multiple GPUs, of simulations of large‐scale networks composed of biologically realistic Hodgkin–Huxley neurons. The algorithm represents each neuron as a CUDA thread, which solves the set of coupled differential equations that model each neuron. Communication among neurons located in different GPUs is coordinated by the CPU. We obtained speedups of 40 for the simulation of 200k neurons that received random external input and speedups of 9 for a network with 200k neurons and 20M neuronal connections, in a single computer with two graphic boards with two GPUs each, when compared with a modern quad‐core CPU. Copyright © 2010 John Wiley & Sons, Ltd.     

Pointwise uniformly convergent numerical treatment for the non-stationary Burger–Huxley equation using grid equidistribution

This article presents a numerical method for solving the singularly perturbed Burger–Huxley equation on a rectangular domain. That is, the highest-order derivative term in the equation is multiplied by a very small parameter. This small parameter is known as the perturbation parameter. When the perturbation parameter specifying the problem tends to zero, the solution of the perturbed problem exhibits layer behaviour in the outflow boundary region. Most conventional methods fail to capture this layer behaviour. For this reason, there is much current interest in the development of a robust numerical method that may handle the difficulties occurring due to the presence of the perturbation parameter and the nonlinearity of the problem. To solve both of these difficulties a numerical method is constructed. The first step in this direction is the discretization of the time variable using Euler's implicit method with a constant time step. This produces a nonlinear stationary singularly perturbed semidiscrete problem class. The problem class is then linearized using the quasilinearization process. This is followed by discretization in space, which uses the standard upwind finite difference operator. An extensive amount of analysis is carried out in order to establish the convergence and stability of the proposed method. Numerical experiments are carried out for model problems to illustrate graphically the theoretical results. The results indicate that the scheme faithfully mimics the dynamics of the differential equation.     

A Computational Analysis to Burgers Huxley Equation

The efficiency of solving computationally partial differential equations can be profoundly highlighted by the creation of precise, higher-order compact numerical scheme that results in truly outstanding accuracy at a given cost. The objective of this article is to develop a highly accurate novel algorithm for two dimensional non-linear Burgers Huxley (BH) equations. The proposed compact numerical scheme is found to be free of superiors approximate oscillations across discontinuities, and in a smooth flow region, it efficiently obtained a high-order accuracy. In particular, two classes of higher-order compact finite difference schemes are taken into account and compared based on their computational economy. The stability and accuracy show that the schemes are unconditionally stable and accurate up to a two-order in time and to six-order in space. Moreover, algorithms and data tables illustrate the scheme efficiency and decisiveness for solving such non-linear coupled system. Efficiency is scaled in terms of L₂ and L_∞ norms, which validate the approximated results with the corresponding analytical solution. The investigation of the stability requirements of the implicit method applied in the algorithm was carried out. Reasonable agreement was constructed under indistinguishable computational conditions. The proposed methods can be implemented for real-world problems, originating in engineering and science.     

An integro-differential generalization and dynamically consistent discretizations of some hyperbolic models with nonlinear damping

This note presents a weak generalization of a time-delayed partial differential equation which, in turn, generalizes the well-known Burger–Fisher and Burgers–Huxley models. In this work, we provide a full discretization which is consistent with the integro-differential equation under consideration. The main analytical result of this note establishes that the discrete temporal rate of change of the discretization yields a consistent approximation to the differential form of the integro-differential equation investigated. Some numerical examples are provided in order to assess the efficiency and effectiveness of our methodology.