Daisuke Furihata

Discrete variational derivative method : a structure-preserving numerical method for partial differential equations

Preface Introduction and Summary of This Book An Introductory Example: the Spinodal Decomposition History Derivation of Dissipative or Conservative Schemes Advanced Topics Target Partial Differential Equations Variational Derivatives First-Order Real-Valued PDEs First-Order Complex-Valued PDEs Systems of First-Order PDEs Second-Order PDEs Discrete Variational Derivative Method Discrete Symbols and Formulas Procedure for First-Order Real-Valued PDEs Procedure for First-Order Complex-Valued PDEs Procedure for Systems of First-Order PDEs Design of Schemes Procedure for Second-Order PDEs Preliminaries on Discrete Functional Analysis Applications Target PDEs Cahn-Hilliard Equation Allen-Cahn Equation Fisher-Kolmogorov Equation Target PDEs Target PDEs Target PDEs Nonlinear Schrodinger Equation Target PDEs Zakharov Equations Target PDEs Other Equations Advanced Topic I: Design of High-Order Schemes Orders of Accuracy of the Schemes Spatially High-Order Schemes Temporally High-Order Schemes: With the Composition Method Temporally High-Order Schemes: With High-Order Discrete Variational Derivatives Advanced Topic II: Design of Linearly-Implicit Schemes Basic Idea for Constructing Linearly-Implicit Schemes Multiple-Points Discrete Variational Derivative Design of Schemes Applications Remark on the Stability of Linearly-Implicit Schemes Advanced Topic III: Further Remarks Solving System of Nonlinear Equations Switch to Galerkin Framework Extension to Non-Rectangular Meshes on D Region A Semi-discrete schemes in space B Proof of Proposition 3.4 Bibliography Index

A stabilization of multistep linearly implicit schemes for dissipative systems

We consider numerical integration of dissipative gradient systems. For such systems, a class of special, stable integrators that strictly maintain dissipation is known, but they generally yield expensive fully implicit schemes, and when the system is large, linearization is indispensable for practical efficiency. However, this can in turn destroy the originally expected stability, and so far no effective principle has been formulated for a stable linearization. In this note, we point out that the behavior of the linearized schemes can be understood from a dynamical systems theory viewpoint and propose a simple principle for a stable linearization.

A LYAPUNOV-TYPE THEOREM FOR DISSIPATIVE NUMERICAL INTEGRATORS WITH ADAPTIVE TIME-STEPPING ∗

The asymptotic behavior of continuous dissipative systems and dissipative numerical integrators with fixed time-stepping can be fully investigated by a Lyapunov-type theorem on continuous and discrete dynamical systems, respectively. However, once adaptive time-stepping is involved, such theories cease to work, and usually the dynamics should be investigated from the past, instead of the standard forward way, such as in terms of pullback attractors. In this paper, we present a different approach---we stick to a forward definition of limit sets and show that still we can establish a Lyapunov-type theorem, which reveals the precise asymptotic behavior of adaptive time-stepping integrators in the presence of a discrete Lyapunov functional.

Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation

We consider the stochastic Cahn--Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension

Some discrete inequalities for central-difference type operators

d\le 3

A new technique to design numerical schemes with weak nonlinearity based on discrete variational derivative method

. We discretize the equation using a standard fi...

Discrete Variational Derivative Method

Discrete versions of basic inequalities in functional analysis such as the Sobolev inequality play key role in theoretical analysis of finite difference schemes. They have been shown for some simple difference operators, but are still left open for general operators, even including the standard central difference operators. In this paper, we propose a systematic approach for deriving such inequalities for a certain class of central-difference type operators. We illustrate the results by giving a generic a priori estimate for certain conservative schemes for the nonlinear Schrodinger equation.

Invariants-preserving integration of the modified Camassa–Holm equation

Generally, discrete variational derivative schemes for nonlinear partial differential equations are nonlinear. The quadratic decomposition of nonlinearity is effective for neither high order polynomial problems nor nonpolynomial ones. Here we propose a new decomposition and new structure preserving schemes based on the decomposition.