Finite difference method

About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.

Flux difference splitting for 1D open channel flow equations

Abstract: SUMMARY An upwind finite difference scheme based on flux difference splitting is presented for the solution of the equations governing unsteady open channel hydraulics. An approximate Jacobian needed for splitting the flux differences is defined that satisfies the conditions required to construct a first-order upwind conservative discretization of the equations. Added limited second-order corrections make the resulting scheme robust and accurate for the computation of all regimes of open channel flow. Some numerical results and comparisons with other classical schemes under exacting conditions are presented.

Analysis of arbitrarily shaped two-dimensional microwave circuits by finite-difference time-domain method

Abstract: A version of the finite-difference time-domain method adapted to the needs of S-matrix calculations of microwave two-dimensional circuits is presented. The analysis is conducted by simulating the wave propagation in the circuit terminated by matched loads and excited by a matched pulse source. Various aspects of the method's accuracy are investigated. Practical computer implementation of the method is discussed, and an example of its application to an arbitrarily shaped microstrip circuit is presented. It is shown that the method in the proposed form is an effective tool of circuit analysis in engineering applications. The method is compared to two other methods used for a similar purpose, namely the contour integral method and the transmission-line matrix method. >

Applications of Lie Groups to Difference Equations

Abstract: Preface Introduction Brief introduction to Lie group analysis of differential equations Preliminaries: Heuristic approach in examples Finite Differences and Transformation Groups in Space of Discrete Variables The Taylor group and finite-difference derivatives Difference analog of the Leibniz rule Invariant difference meshes Transformations preserving the geometric meaning of finite-difference derivatives Newton's group and Lagrange's formula Commutation properties and factorization of group operators on uniform difference meshes Finite-difference integration and prolongation of the mesh space to nonlocal variables Change of variables in the mesh space Invariance of Finite-Difference Models An invariance criterion for finite-difference equations on the difference mesh Symmetry preservation in difference modeling: Method of finite-difference invariants Examples of construction of difference models preserving the symmetry of the original continuous models Invariant Difference Models of Ordinary Differential Equations First-order invariant difference equations and lattices Invariant second-order difference equations and lattices Invariant Difference Models of Partial Differential Equations Symmetry preserving difference schemes for the nonlinear heat equation with a source Symmetry preserving difference schemes for the linear heat equation Invariant difference models for the Burgers equation Invariant difference model of the heat equation with heat flux relaxation Invariant difference model of the Korteweg-de Vries equation Invariant difference model of the nonlinear Shrodinger equation Combined Mathematical Models and Some Generalizations Second-order ordinary delay differential equations Partial delay differential equations Symmetry of differential-difference equations Lagrangian Formalism for Difference Equations Discrete representation of Euler's operator Criterion for the invariance of difference functionals Invariance of difference Euler equations Variation of difference functional and quasi-extremal equations Invariance of global extremal equations and properties of quasiextremal equations Conservation laws for difference equations Noether-type identities and difference analog of Noether's theorem Necessary and sufficient conditions for global extremal equations to be invariant Applications of Lagrangian formalism to second-order difference equations Moving mesh schemes for the nonlinear Shrodinger equation Hamiltonian Formalism for Difference Equations: Symmetries and First Integrals Discrete Legendre transform Variational statement of the difference Hamiltonian equations Symplecticity of difference Hamiltonian equations Invariance of the Hamiltonian action Difference Hamiltonian identity and Noether-type theorem for difference Hamiltonian equations Invariance of difference Hamiltonian equations Examples Discrete Representation of Ordinary Differential Equations with Symmetries The discrete representation of ODE as a series Three-point exact schemes for nonlinear ODE Bibliography Index

Gauss Point Mass Lumping Schemes for Maxwell's Equations

Abstract: Finite element and finite difference methods for approximating the Maxwell system propagate numerical waves with slightly incorrect velocities, and this results in phase error in the computed solution. Indeed this error limits the type of problem that can be solved, because phase error accumulates during the computation and eventually destroys the solution. Here we propose a family of mass-lumped finite element schemes using edge elements. We emphasize in particular linear elements that are equivalent to the standard Yee FDTD scheme, and cubic elements that have superior phase accuracy. We prove theorems that allow us to perform a dispersion analysis of the two common families of edge elements on rectilinear grids. A result of this analysis is to provide some justification for the choice of the particular family we use. We also provide a limited selection of numerical results that show the efficiency of our scheme. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 63–88, 1998