Related papers: A novel energy-conserving scheme for eight-dimensi…
In this paper, we apply the coordinate increment discrete gradient (CIDG) method to solve the Lorentz force system which can be written as a non-canonical Hamiltonian system. Then we can obtain a new energy-preserving CIDG-I method for the…
Runge-Kutta methods are affine equivariant: applying a method before or after an affine change of variables yields the same numerical trajectory. However, for some applications, one would like to perform numerical integration after a…
Recently, a whole class of evergy-preserving integrators has been derived for the numerical solution of Hamiltonian problems. In the mainstream of this research, we have defined a new family of symplectic integrators depending on a real…
Conservation properties of iterative methods applied to implicit finite volume discretizations of nonlinear conservation laws are analyzed. It is shown that any consistent multistep or Runge-Kutta method is globally conservative. Further,…
Variational integrators have traditionally been constructed from the perspective of Lagrangian mechanics, but there have been recent efforts to adopt discrete variational approaches to the symplectic discretization of Hamiltonian mechanics…
Coherent or exact equations of motion for a post-Newtonian Lagrangian formalism are the Euler-Lagrange equations without any terms truncated. They naturally conserve energy {and} angular momentum. Doubling the phase-space variables of…
An energetically balanced, implicit integrator for non-hydrostatic vertical atmospheric dynamics on the sphere is presented. The integrator allows for the exact balance of energy exchanges in space and time for vertical atmospheric motions…
One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants, among which the Hamiltonian function itself. For example, it is well known that classical symplectic…
Using Suzuki-Trotter decompositions of exponential operators we describe new algorithms for the numerical integration of the equations of motion for classical spin systems. These techniques conserve spin length exactly and, in special…
Dynamic systems have a fundamental relevance in the description of physical phenomena. The search for more accurate and faster numerical integration methods for the resolution of such systems is, therefore, an important topic of research.…
Hamiltonian systems are one of the most important class of dynamical systems with a geometric structure called symplecticity and the numerical algorithms which can preserve such geometric structure are of interest. In this article we study…
Studying single-particle dynamics over many periods of oscillations is a well-understood problem solved using symplectic integration. Such integration schemes derive their update sequence from an approximate Hamiltonian, guaranteeing that…
Due to its remarkable energy compaction properties, the discrete cosine transform (DCT) is employed in a multitude of compression standards, such as JPEG and H.265/HEVC. Several low-complexity integer approximations for the DCT have been…
Symplectic integration methods based on operator splitting are well established in many branches of science. For Hamiltonian systems which split in more than two parts, symplectic methods of higher order have been studied in detail only for…
In this paper, we present and study discontinuous Galerkin (DG) methods for one-dimensional multi-symplectic Hamiltonian partial differential equations. We particularly focus on semi-discrete schemes with spatial discretization only, and…
We consider numerical simulation of the isotropic elastic wave equations arising from seismic applications with non-trivial land topography. The more flexible finite element method is applied to the shallow region of the simulation domain…
There is a growing interest in the conservation of invariants when numerically solving a system of ordinary differential equations. Methods that exactly preserve these quantities in time are known as geometric integrators. In this paper we…
Reduced basis methods are popular for approximately solving large and complex systems of differential equations. However, conventional reduced basis methods do not generally preserve conservation laws and symmetries of the full order model.…
We present linearly implicit methods that preserve discrete approximations to local and global energy conservation laws for multi-symplectic PDEs with cubic invariants. The methods are tested on the one-dimensional Korteweg-de Vries…
We analyze the wave equation in mixed form, with periodic and/or Dirichlet homogeneous boundary conditions, and nonconstant coefficients that depend on the spatial variable. For the discretization, the weak form of the second equation is…