Finite difference

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

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

Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation

Abstract: We present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on the L1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.

Optimal Error Estimates for the Fully Discrete Interior Penalty DG Method for the Wave Equation

Abstract: In Grote et al. (SIAM J. Numer. Anal., 44:2408---2431, 2006) a symmetric interior penalty discontinuous Galerkin (DG) method was presented for the time-dependent wave equation. In particular, optimal a-priori error bounds in the energy norm and the L 2-norm were derived for the semi-discrete formulation. Here the error analysis is extended to the fully discrete numerical scheme, when a centered second-order finite difference approximation ("leap-frog" scheme) is used for the time discretization. For sufficiently smooth solutions, the maximal error in the L 2-norm error over a finite time interval converges optimally as O(h p+1+Δt 2), where p denotes the polynomial degree, h the mesh size, and Δt the time step.