Discrete variational derivative method—A structure-preserving numerical method for partial differential equations

Furihata, Daisuke; Matsuo, Takayasu

doi:10.1090/suga/435

Cited by 78 publications

(123 citation statements)

References 35 publications

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“…The operator DW is usually referred to as discrete variational derivative. The use of discrete variational derivatives is ubiquitous in the design of exact energy preserving schemes and symplectic methods; see e.g., (Furihata and Matsuo, 2011;Simo et al, 1992;Romero, 2009). …”

Section: Secant Methodsmentioning

confidence: 99%

Computational Phase‐Field Modeling

Gómez

Zee

2017

Encyclopedia of Computational Mechanics Second Edition

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Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community.

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Section: Secant Methodsmentioning

confidence: 99%

Computational Phase‐Field Modeling

Gómez

Zee

2017

Encyclopedia of Computational Mechanics Second Edition

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“…which is a typical choice in [26]; however, we can also use, for example, the central difference and define the Hamiltonian as…”

Section: Remarkmentioning

confidence: 99%

“…A similar method for PDEs also exists, which is called the discrete variational derivative method (e.g. [22][23][24]26]). Other examples are symplectic integrators, which are numerical methods that preserve the symplecticity of the Hamiltonian flow in the discrete setup.…”

Section: Introductionmentioning

confidence: 99%

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Geometric-integration tools for the simulation of musical sounds

Ishikawa

Michels

Yaguchi

2018

Japan J. Indust. Appl. Math.

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During the last decade, much attention has been given to sound rendering and the simulation of acoustic phenomena by solving appropriate models described by Hamiltonian partial differential equations. In this contribution, we introduce a procedure to develop appropriate tools inspired from geometric integration in order to simulate musical sounds. Geometric integrators are numerical integrators of excellent quality that are designed exclusively for Hamiltonian ordinary differential equations. The introduced procedure is a combination of two techniques in geometric integration: the semi-discretization method by Celledoni et al. (J Comput Phys 231:6770-6789, 2012) and symplectic partitioned Runge-Kutta methods. This combination turns out to be a right procedure that derives numerical schemes that are effective and suitable for computation of musical sounds. By using this procedure we derive a series of explicit integration algorithms for a simple model describing piano sounds as a representative example for virtual instruments. We demonstrate the advantage of the numerical methods by evaluating a variety of numerical test cases.

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“…The discrete variational derivative method (DVDM) is a class of structure-preserving methods, proposed by Furihata and Matsuo (2010), and it can retain the conservation/dissipation properties of the original equations. Up to now, the DVDM has been applied to many conservative or dissipative partial differential equations (PDEs).…”

Section: Introductionmentioning

confidence: 99%

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Yan

Zheng

2017

International Journal of Applied Mathematics and Computer Science

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The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg-de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg-de Vries equation.

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Discrete variational derivative method—A structure-preserving numerical method for partial differential equations

Cited by 78 publications

References 35 publications

Computational Phase‐Field Modeling

Computational Phase‐Field Modeling

Geometric-integration tools for the simulation of musical sounds

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

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