The Experts below are selected from a list of 216 Experts worldwide ranked by ideXlab platform
Takayasu Matsuo - One of the best experts on this subject based on the ideXlab platform.
-
Discrete Variational Derivative Methods for the EPDiff Equation
arXiv: Analysis of PDEs, 2016Co-Authors: Stig Larsson, Takayasu Matsuo, Klas Modin, Matteo MolteniAbstract:The aim of this paper is the derivation of structure preserving schemes for the solution of the EPDiff equation, with particular emphasis on the two dimensional case. We develop three different schemes based on the Discrete Variational Derivative Method (DVDM) on a rectangular domain discretized with a regular, structured, orthogonal grid. We present numerical experiments to support our claims: we investigate the preservation of energy and linear momenta, the reversibility, and the empirical convergence of the schemes. The quality of our schemes is finally tested by simulating the interaction of singular wave fronts.
-
A novel discrete Variational Derivative method using ``average-difference methods''
JSIAM Letters, 2016Co-Authors: Daisuke Furihata, Shun Sato, Takayasu MatsuoAbstract:We consider structure-preserving methods for conservative systems, which rigorously replicate the conservation property yielding better numerical solutions. There, corresponding to the skew-symmetry of the differential operator, that of difference operators is essential to the discrete conservation law. Unfortunately, however, when we employ the standard central difference operator, the simplest one, the numerical solutions often suffer from undesirable spatial oscillations. In this letter, we propose a novel "average-difference method," which is tougher against such oscillations, and combine it with an existing conservative method. Theoretical and numerical analysis in the linear case show the superiority of the proposed method.
-
An alternating discrete Variational Derivative method
2012Co-Authors: Takayasu Matsuo, Hiroaki KuramaeAbstract:A new “alternating” computation approach for dissipative or conservative partial differential equations is presented. The method is obtained by extending an existing “discrete Variational Derivative method,” which is a method for designing special numerical schemes that retain the dissipation or conservation properties of the original partial differential equations. It is also shown that the alternating approach can be further combined with an existing linearization technique to construct highly efficient numerical schemes. A numerical example illustrating the high efficiency is given.
-
The discrete Variational Derivative method based on discrete differential forms
Journal of Computational Physics, 2012Co-Authors: Takaharu Yaguchi, Takayasu Matsuo, Masaaki SugiharaAbstract:As is well known, for PDEs that enjoy a conservation or dissipation property, numerical schemes that inherit this property are often advantageous in that the schemes are fairly stable and give qualitatively better numerical solutions in practice. Lately, Furihata and Matsuo have developed the so-called "discrete Variational Derivative method" that automatically constructs energy preserving or dissipative finite difference schemes. Although this method was originally developed on uniform meshes, the use of non-uniform meshes is of importance for multi-dimensional problems. On the other hand, the theories of discrete differential forms have received much attention recently. These theories provide a discrete analogue of the vector calculus on general meshes. In this paper, we show that the discrete Variational Derivative method and the discrete differential forms by Bochev and Hyman can be combined. Applications to the Cahn-Hilliard equation and the Klein-Gordon equation on triangular meshes are provided as demonstrations. We also show that the schemes for these equations are H1-stable under some assumptions. In particular, one for the nonlinear Klein-Gordon equation is obtained by combination of the energy conservation property and the discrete Poincare inequality, which are the temporal and spacial structures that are preserved by the above methods.
-
discrete Variational Derivative method a structure preserving numerical method for partial differential equations
2010Co-Authors: Daisuke Furihata, Takayasu MatsuoAbstract: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
Daisuke Furihata - One of the best experts on this subject based on the ideXlab platform.
-
A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition.
arXiv: Numerical Analysis, 2020Co-Authors: Makoto Okumura, Takeshi Fukao, Daisuke Furihata, Shuji YoshikawaAbstract:We propose a structure-preserving finite difference scheme for the Cahn-Hilliard equation with a dynamic boundary condition using the discrete Variational Derivative method (DVDM). In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal Derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme by Fukao-Yoshikawa-Wada (Commun. Pure Appl. Anal. 16 (2017), 1915-1938) is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for the proposed scheme. Computation examples demonstrate the effectiveness of the proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.
-
A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition
Discrete & Continuous Dynamical Systems - A, 2020Co-Authors: Makoto Okumura, Daisuke FurihataAbstract:We propose a structure-preserving finite difference scheme for the Allen–Cahn equation with a dynamic boundary condition using the discrete Variational Derivative method [ 9 ]. In this method, how to discretize the energy which characterizes the equation is essential. Modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a central difference operator as an approximation of an outward normal Derivative on the boundary condition in the scheme. We show the stability and the existence and uniqueness of the solution for the proposed scheme. Also, we give the error estimate for the scheme. Numerical experiments demonstrate the effectiveness of the proposed scheme. Besides, through numerical experiments, we confirm that the long-time behavior of the solution under a dynamic boundary condition may differ from that under the Neumann boundary condition.
-
a structure preserving scheme for the allen cahn equation with a dynamic boundary condition
Discrete and Continuous Dynamical Systems, 2020Co-Authors: Makoto Okumura, Daisuke FurihataAbstract:We propose a structure-preserving finite difference scheme for the Allen–Cahn equation with a dynamic boundary condition using the discrete Variational Derivative method [ 9 ]. In this method, how to discretize the energy which characterizes the equation is essential. Modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a central difference operator as an approximation of an outward normal Derivative on the boundary condition in the scheme. We show the stability and the existence and uniqueness of the solution for the proposed scheme. Also, we give the error estimate for the scheme. Numerical experiments demonstrate the effectiveness of the proposed scheme. Besides, through numerical experiments, we confirm that the long-time behavior of the solution under a dynamic boundary condition may differ from that under the Neumann boundary condition.
-
A novel discrete Variational Derivative method using ``average-difference methods''
JSIAM Letters, 2016Co-Authors: Daisuke Furihata, Shun Sato, Takayasu MatsuoAbstract:We consider structure-preserving methods for conservative systems, which rigorously replicate the conservation property yielding better numerical solutions. There, corresponding to the skew-symmetry of the differential operator, that of difference operators is essential to the discrete conservation law. Unfortunately, however, when we employ the standard central difference operator, the simplest one, the numerical solutions often suffer from undesirable spatial oscillations. In this letter, we propose a novel "average-difference method," which is tougher against such oscillations, and combine it with an existing conservative method. Theoretical and numerical analysis in the linear case show the superiority of the proposed method.
-
A new technique to design numerical schemes with weak nonlinearity based on discrete Variational Derivative method
2012Co-Authors: Daisuke FurihataAbstract: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.
Yukihito Suzuki - One of the best experts on this subject based on the ideXlab platform.
-
bracket formulations and energy and helicity preserving numerical methods for the three dimensional vorticity equation
Computer Methods in Applied Mechanics and Engineering, 2017Co-Authors: Yukihito SuzukiAbstract:Abstract The vorticity equation for three-dimensional viscous incompressible fluid flows is formulated within different bracket formalisms using the Poisson or Nambu bracket together with a dissipative bracket. The budgets of kinetic energy, helicity, and enstrophy derived from the bracket formulations are properly inherited by the finite difference equations obtained by invoking the discrete Variational Derivative method combined with the mimetic finite difference method. In particular, energy and helicity are conserved precisely in inviscid flow computations. The energy and enstrophy dissipate properly owing to viscosity in viscous flow computations, and the enstrophy is appropriately produced by the vortex stretching effect in both inviscid and viscous flow computations. The relationships between the stream function, velocity, and vorticity as well as the solenoidal conditions on the velocity and vorticity fields are also inherited. Numerical experiments on a periodic array of rolls that permits analytical solutions have been done to examine the properties and usefulness of the proposed method.
-
GENERIC formalism and discrete Variational Derivative method for the two-dimensional vorticity equation
Journal of Computational and Applied Mathematics, 2016Co-Authors: Yukihito Suzuki, Masashi OhnawaAbstract:The vorticity equation for two-dimensional incompressible viscous flows is formulated within the GENERIC formalism for non-equilibrium thermodynamics. The laws of conservation of energy and increasing entropy derived from the GENERIC formulation are properly inherited by the finite difference equations obtained by invoking the discrete Variational Derivative method. The law of increasing entropy corresponds to the dissipation of enstrophy for the vorticity equation. Some numerical experiments have been done to examine the usefulness of the proposed method.
Rehana Naz - One of the best experts on this subject based on the ideXlab platform.
-
Conservation laws for a complexly coupled KdV system, coupled Burgers’ system and Drinfeld-Sokolov-Wilson system via multiplier approach
Communications in Nonlinear Science and Numerical Simulation, 2010Co-Authors: Rehana NazAbstract:Abstract The multiplier approach (Variational Derivative method) is used to derive the conservation laws for some nonlinear systems of partial differential equations. Firstly, the multipliers (characteristics) are computed and then conserved vectors are obtained for the each multiplier. Examples of the third-order complexly coupled KdV system, second-order coupled Burgers’ system and third-order Drinfeld–Sokolov–Wilson system are considered. For all three systems the local conservation laws are established by utilizing the multiplier approach.
-
Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation
Nonlinear Analysis: Real World Applications, 2009Co-Authors: Rehana Naz, I. Naeem, Shirley AbelmanAbstract:Abstract In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The Variational Derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the Variational Derivative approach yields two multipliers; thus two conserved vectors are obtained.
-
Wall Jet on a Hemi-spherical Shell: Conserved Quantities and Group Invariant Solution
2009Co-Authors: I. Naeem, Rehana NazAbstract:The Variational Derivative approach is used to derive the conservation laws for the system of equations for the velocity components as well as for the third-order partial differen- tial equation for the stream function governing flow outside and inside a hemi-spherical shell. One of the conserved vectors for the stream function equation is used to derive the conserved quantity for the wall jet. A symmetry is associated with that conserved vector and then this symmetry is used to generate the group invariant solution for the wall jet formed on the outside or inside of the shell.
Salomon R. Billeter - One of the best experts on this subject based on the ideXlab platform.
-
Analytic second Variational Derivative of the exchange-correlation functional
Physical Review B, 2004Co-Authors: Daniel Egli, Salomon R. BilleterAbstract:A general analytic expression for the second Variational Derivative of gradient-corrected exchange-correlation energy functionals is derived, and the terms for the widely used Becke/Perdew, Becke/Lee-Yang-Parr, and Perdew-Burke-Ernzerhof exchange-correlation functionals are given. These analytic Derivatives can be used for all applications employing linear-response theory or time-dependent density-functional theory. Calculations are performed in a plane-wave scheme and shown to be numerically more stable, more accurate, and computationally less costly than the most widely used finite-difference scheme.
