822 research outputs found
On discretely entropy conservative and entropy stable discontinuous Galerkin methods
High order methods based on diagonal-norm summation by parts operators can be
shown to satisfy a discrete conservation or dissipation of entropy for
nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as
nodal discontinuous Galerkin methods with diagonal mass matrices. In this work,
we describe how use flux differencing, quadrature-based projections, and
SBP-like operators to construct discretely entropy conservative schemes for DG
methods under more arbitrary choices of volume and surface quadrature rules.
The resulting methods are semi-discretely entropy conservative or entropy
stable with respect to the volume quadrature rule used. Numerical experiments
confirm the stability and high order accuracy of the proposed methods for the
compressible Euler equations in one and two dimensions
Speed selection for traveling-wave solutions to the diffusion-reaction equation with cubic reaction term and Burgers nonlinear convection
The problem of traveling wave (TW) speed selection for solutions to a generalized Murray-Burgers-KPP-Fisher parabolic equation with a strictly positive cubic reaction term is considered theoretically and the initial boundary value problem is numerically solved in order to support obtained analytical results. Depending on the magnitude of a parameter inherent in the reaction term (i) the term is either a concave function or a function with the inflection point and (ii) transition from pulled to pushed TW solution occurs due to interplay of two nonlinear terms; the reaction term and the Burgers convection term. Explicit pushed TW solutions are derived. It is shown that physically observable TW solutions, i.e., solutions obtained by solving the initial boundary value problem with a sufficiently steep initial condition, can be determined by seeking the TW solution characterized by the maximum decay rate at its leading edge. In the Appendix, the developed approach is applied to a non-linear diffusion-reaction equation that is widely used to model premixed turbulent combustion
Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms
Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear time-delayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters , and the resulting equations together with the two-point boundary conditions constitute a system of ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms
Homotopy Asymptotic Method and Its Application
As we all know, perturbation theory is closely related to methods used in the numerical analysis fields. In this chapter, we focus on introducing two homotopy asymptotic methods and their applications. In order to search for analytical approximate solutions of two types of typical nonlinear partial differential equations by using the famous homotopy analysis method (HAM) and the homotopy perturbation method (HPM), we consider these two systems including the generalized perturbed Kortewerg-de Vries-Burgers equation and the generalized perturbed nonlinear Schrödinger equation (GPNLS). The approximate solution with arbitrary degree of accuracy for these two equations is researched, and the efficiency, accuracy and convergence of the approximate solution are also discussed
Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation
This paper investigates the solitary wave solutions of the generalized Rosenau–Korteweg-de Vries-regularized-long wave equation. This model is obtained by coupling the Rosenau–Korteweg-de Vries and Rosenau-regularized-long wave equations. The solution of the equation is approximated by a local meshless technique called radial basis function (RBF) and the finite-difference (FD) method. The association of the two techniques leads to a meshless algorithm that does not requires the linearization of the nonlinear terms. First, the partial differential equation is transformed into a system of ordinary differential equations (ODEs) using radial kernels. Then, the ODE system is solved by means of an ODE solver of higher-order. It is shown that the proposed method is stable. In order to illustrate the validity and the efficiency of the technique, five problems are tested and the results compared with those provided by other schemes.info:eu-repo/semantics/publishedVersio
Disorder-Induced Depinning Transition
The competition in the pinning of a directed polymer by a columnar pin and a
background of random point impurities is investigated systematically using the
renormalization group method. With the aid of the mapping to the noisy-Burgers'
equation and the use of the mode-coupling method, the directed polymer is shown
to be marginally localized to an arbitrary weak columnar pin in 1+1 dimensions.
This weak localization effect is attributed to the existence of large scale,
nearly degenerate optimal paths of the randomly pinned directed polymer. The
critical behavior of the depinning transition above 1+1 dimensions is obtained
via an -expansion.Comment: 47 pages in revtex; postscript files of 6 figures include
Finite difference method for numerical solution of a generalized burgers-huxley equation
There are many applications of the generalized Burgers-Huxley equation which is a form of nonlinear Partial Differential Equation such as in the work of physicist which can effectively models the interaction between reaction mechanisms, convection effects and diffusion transports. This study investigates on the implementation of numerical method for solving the generalized Burgers-Huxley equation. The method is known as the Finite Difference Method which can be employed using several approaches and this work focuses on the Explicit Method, the Modified Local Crank-Nicolson (MLCN) Method and Nonstandard Finite Difference Schemes (NFDS). In order to use the NFDS, due to a lack of boundary condition provided in the problem, this research used the Forward Time Central Space (FTCS) Method to approximate the first step in time. Thomas Algorithm was applied for the methods that lead to a system of linear equation. Computer codes are provided for these methods using the MATLAB software. The results obtained are compared among the three methods with the exact solution for determining their accuracy. Results shows that NFDS has the lowest relative error and one of the best way among these three methods in order to solve the generalized Burgers-Huxley equations
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