Discrete variational derivative method: A by Furihata D., Matsuo T.

By Furihata D., Matsuo T.

Nonlinear Partial Differential Equations (PDEs) became more and more vital within the description of actual phenomena. not like traditional Differential Equations, PDEs can be utilized to successfully version multidimensional platforms. The tools recommend in Discrete Variational spinoff technique be aware of a brand new category of "structure-preserving numerical equations" which improves the qualitative behaviour of the PDE strategies and allows strong computing. The authors have additionally taken care to provide their equipment in an available demeanour, because of this the e-book should be precious to engineers and physicists with a uncomplicated wisdom of numerical research. subject matters mentioned contain: "Conservative" equations resembling the Korteweg–de Vries equation (shallow water waves) and the nonlinear Schrödinger equation (optical waves) "Dissipative" equations equivalent to the Cahn–Hilliard equation (some section separation phenomena) and the Newell-Whitehead equation (two-dimensional Bénard convection movement) layout of spatially and temporally high-order schemas layout of linearly-implicit schemas fixing structures of nonlinear equations utilizing numerical Newton process libraries

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Extra resources for Discrete variational derivative method: A structure-preserving numerical method for PDE

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Let us consider next the cases where utt is concerned; we call such PDEs secondorder. Let u(x, t) be a real-valued function, and suppose that a real-valued function G(u, ux ) is given. We here consider the PDE of the form ∂2u δG =− , ∂t2 δu x ∈ (0, L), t > 0. 75) 28 Discrete Variational Derivative Method provided some appropriate boundary conditions are given. This class of PDEs includes, for example, the linear wave equation ∂2u ∂2u = , ∂t2 ∂x2 G(u, ux ) = (ux )2 . 75) now includes not only G(u, ux ) but also ut in its integrand.

Let us consider next the cases where utt is concerned; we call such PDEs secondorder. Let u(x, t) be a real-valued function, and suppose that a real-valued function G(u, ux ) is given. We here consider the PDE of the form ∂2u δG =− , ∂t2 δu x ∈ (0, L), t > 0. 75) 28 Discrete Variational Derivative Method provided some appropriate boundary conditions are given. This class of PDEs includes, for example, the linear wave equation ∂2u ∂2u = , ∂t2 ∂x2 G(u, ux ) = (ux )2 . 75) now includes not only G(u, ux ) but also ut in its integrand.

The periodic boundary condition is assumed for simplicity so that we can forget this issue. Rewriting the procedure of the standard discrete variational derivative 〈1〉,2p method with the high-order difference operator δk , we obtain a new procedure for a spatially higher-order discrete variational derivative method. Let us see this in the case of the first-order real-valued PDEs: ∂u δG =− , ∂t δu x ∈ (0, L), t > 0. 23) with G(u, ux ) = (ux )2 /2. [step 1d ] Defining a discrete energy 〈1〉,2p By simply replacing u in G(u, ux ) with Uk (m) , and ux with δk Uk (m) , we (m) obtain a discrete energy Gd (U ).

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