Related papers: SIMD-vectorized implicit symplectic integrators ca…
In current research, we analyse dissipation and dispersion characteristics of most accurate two and three stage Gauss-Legendre implicit Runge-Kutta (R-K) methods. These methods, known for their $A$-stability and immense accuracy, are…
Implicit integration of the viscous term can significantly improve performance in computational fluid dynamics for highly viscous fluids such as lava. We show improvements over our previous proposal for semi-implicit viscous integration in…
In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge-Kutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems,…
We investigate the performance of various methods of symplectic integration, which are based on two part splitting of the integration operator, for the numerical integration of multidimensional Hamiltonian systems. We implement these…
Dependable numerical results from long-time simulations require stable numerical integration schemes. For Hamiltonian systems, this is achieved with symplectic integrators, which conserve the symplectic condition and exactly solve for the…
We introduce a class of fourth order symplectic algorithms that are ideal for doing long time integration of gravitational few-body problems. These algorithms have only positive time steps, but require computing the force gradient in…
This paper considers the numerical integration of semilinear evolution PDEs using the high order linearly implicit methods developped in a previous paper in the ODE setting. These methods use a collocation Runge--Kutta method as a basis,…
Most numerical integration algorithms are not designed specifically for Hamiltonian systems and do not respect their characteristic properties, which include the preservation of phase space volume with time. This can lead to spurious…
SSE (streaming SIMD extensions) and AVX (advanced vector extensions) are SIMD (single instruction multiple data streams) instruction sets supported by recent CPUs manufactured in Intel and AMD. This SIMD programming allows parallel…
Finite element discretization of time dependent problems also require effective time-stepping schemes. While implicit Runge-Kutta methods provide favorable accuracy and stability problems, they give rise to large and complicated systems of…
Efficient ordinary differential equation solvers for chemical kinetics must take into account the available thread and instruction-level parallelism of the underlying hardware, especially on many-core coprocessors, as well as the numerical…
In this work we propose a new numerical approach to distinguish between regular and chaotic orbits in Hamiltonian systems, based on the simultaneous integration of both the orbit and the deviation vectors using a symplectic scheme, hereby…
Matrix-multiplication units (MXUs) are now prevalent in every computing platform. The key attribute that makes MXUs so successful is the semiring structure, which allows tiling for both parallelism and data reuse. Nonetheless,…
We consider the numerical integration of the matrix Hill's equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, the Hill's equations originate from a Hamiltonian…
Here we present Symplectically Integrated Symbolic Regression (SISR), a novel technique for learning physical governing equations from data. SISR employs a deep symbolic regression approach, using a multi-layer LSTM-RNN with mutation to…
Isospectral flows appear in a variety of applications, e.g. the Toda lattice in solid state physics or in discrete models for two-dimensional hydrodynamics, with the isospectral property often corresponding to mathematically or physically…
The purpose of this paper is presenting a theoretical basis for the study of $\omega$-Hamiltonian vector fields in a more general approach than the classical one. We introduce the concepts of $\omega$-symplectic group and…
We compare three approaches for structure preserving numerical integration of isospectral flows on quadratic Lie algebras. Such flows originate from Hamiltonian dynamics on the cotangent bundle of the Lie group. It is known, via discrete…
In this paper, we study symplectic integration of canonical Hamiltonian systems with Jacobi polynomials. The relevant theoretical results of continuous-stage Runge-Kutta methods are revisited firstly and then symplectic methods with Jacobi…
As is known that various dynamical systems including all Hamiltonian systems preserve volume in phase space. This qualitative geometrical property of the analytical solution should be respected in the sense of Geometric Integration. This…