Abstract
Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler–Lagrange equations) and the conservation laws (Nœther’s theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.
Highlights
With the increasing performance of computers, one might think that it is not worth to design completely new algorithms to solve numerically basic problems in mechanics
Domain discretization and elements of algebraic topology Consider a differential equation defined on a nx-dimensional spatial domain M in Rn, n ≥ nx, written within the exterior calculus framework
With this article, we aimed at raising the awarness among the readers, and among high-performance computing specialists, about the importance of the geometric structure of their equations
Summary
With the increasing performance of computers (speed, parallel processing, storage capacity, ...), one might think that it is not worth to design completely new algorithms to solve numerically basic problems in mechanics. As for it, RK4sym reproduces the quasi-periodicity of the solution thanks to its symplecticity property These results show the importance of the preservation of the equation’s structure when simulating Hamiltonian ODEs. In order to extend symplectic integrators to PDEs, multisymplectic geometry will be introduced in the subsection. It is first order in time and second in x Any solution of this equation verifies the following discrete multisymplectic conservation law. A multisymplectic Gauss–Legendre discretization scheme of this equation is a combination of space and time Runge–Kutta integrators with (sv + sw) stages, defined with the intermediate variables (Ui, Vi)i=1,...,sv and (Ui, Wi)i=1,...,sw as follows: sw uni+1 = uni + x bi xUin, sw win+1 = win + x bi xWin i=1 i=1 where Uin = uni +. We recall the basis of DEC and present some test cases
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