Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

Abstract: In this study, with the aid of Wolfram Mathematica 11, the modified exp ( − Ω ( η ) ) -expansion function method is used in constructing some new analytical solutions with novel structure such as the trigonometric and hyperbolic function solutions to the well-known nonlinear evolutionary equation, namely; the two-component second order KdV evolutionary system. Second, the finite forward difference method is used in analyzing the numerical behavior of this equation. We consider equation (6.5) and (6.6) for the numerical analysis. We examine the stability of the two-component second order KdV evolutionary system with the finite forward difference method by using the Fourier-Von Neumann analysis. We check the accuracy of the finite forward difference method with the help of L 2 and L ∞ norm error. We present the comparison between the exact and numerical solutions of the two-component second order KdV evolutionary system obtained in this article which and support with graphics plot. We observed that the modified exp ( − Ω ( η ) ) -expansion function method is a powerful approach for finding abundant solutions to various nonlinear models and also finite forward difference method is efficient for examining numerical behavior of different nonlinear models.

Finite element approximation to two-dimensional sine-Gordon solitons

Abstract: The paper presents a finite element algorithm for the numerical solution of the sine-Gordon equation in two spatial dimensions, as it arises, for example, in rectangular large-area Josephson junctions. The dispersive nonlinear partial differential equation of the system allows for soliton-type solutions, an ubiquitous phenomenon in a large variety of physical problems. A semidiscrete Galerkin approach based on simple four-noded bilinear finite elements in combination with a generalized Newmark integration scheme is used throughout the paper and is tested in a variety of cases. Comparisons with finite difference solutions show the superior performance of the proposed algorithm leading to very accurate, numerically stable and physically consistent solitary wave solutions. The results support the confidence in the present numerical model which should be capable to treat also more complex situations involving soliton-type interactions.

Inviscid transonic flow over airfoils

Abstract: A procedure is presented to calculate steady supercritical planar flows over lifting airfoils using an unsteady approach, where the steady flow is obtained as the asymptotic flow for large times. The unsteady flow is generated by impulsively imposing the airfoil boundary condition in an initially uniform flow. The resulting flow is calculated by a finite difference analogue to the unsteady Euler equations using a diffusing second-order difference scheme. Here an artificial viscosity appears by which shock waves acquire a steep profile. The procedure is used to calculate the flows over one of the nonlifting symmetrical shockless profiles derived by Nieuwland, using the hodograph method, and over a lifting NAG A 64A-410 profile. Results agree well with experiments, with local differences accountable by a BusemannGuderley instability in the first case, and by viscous effects in the second case.

The Orthogonal Decomposition Theorems for Mimetic Finite Difference Methods

Abstract: Accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus provide reliable finite difference methods for approximating the solutions to a wide class of partial differential equations. These methods mimic many fundamental properties of the underlying physical problem including conservation laws, symmetries in the solution, and the nondivergence of particular vector fields (i.e., they are divergence free) and should satisfy a discrete version of the orthogonal decomposition theorem. This theorem plays a fundamental role in the theory of generalized solutions and in the numerical solution of physical models, including the Navier--Stokes equations and in electrodynamics. We are deriving mimetic finite difference approximations of the divergence, gradient, and curl that satisfy discrete analogs of the integral identities satisfied by the differential operators. We first define the natural discrete divergence, gradient, and curl operators based on coordinate invariant definitions, such as Gauss's theorem, for the divergence. Next we use the formal adjoints of these natural operators to derive compatible divergence, gradient, and curl operators with complementary domains and ranges of values. In this paper we prove that these operators satisfy discrete analogs of the orthogonal decomposition theorem and demonstrate how a discrete vector can be decomposed into two orthogonal vectors in a unique way, satisfying a discrete analog of the formula $\vec{A} = \ggrad \, \varphi + \curl \, \vec{B}$. We also present a numerical example to illustrate the numerical procedure and calculate the convergence rate of the method for a spiral vector field.