2,644 research outputs found
Explicit exactly energy-conserving methods for Hamiltonian systems
For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each time step, or are explicit, but where the exact conservation property depends on the exact evaluation of an integral in continuous time. Under further restrictions, namely that the potential energy contribution to the Hamiltonian is non-negative, newer techniques based on invariant energy quadratisation allow for exact numerical energy conservation and yield linearly implicit updates, requiring only the solution of a linear system at each time step. In this article, it is shown that, for a general class of Hamiltonian systems, and under the non-negativity condition on potential energy, it is possible to arrive at a fully explicit method that exactly conserves numerical energy. Furthermore, such methods are unconditionally stable, and are of comparable computational cost to the very simplest integration methods (such as Störmer-Verlet). A variant of this scheme leading to a conditionally-stable method is also presented, and follows from a splitting of the potential energy. Various numerical results are presented, in the case of the classic test problem of Fermi, Pasta and Ulam and for nonlinear systems of partial differential equations, including those describing high amplitude vibration of strings and plates
Numerical Study of the Two-Species Vlasov-Amp\`{e}re System: Energy-Conserving Schemes and the Current-Driven Ion-Acoustic Instability
In this paper, we propose energy-conserving Eulerian solvers for the
two-species Vlasov-Amp\`{e}re (VA) system and apply the methods to simulate
current-driven ion-acoustic instability. The algorithm is generalized from our
previous work for the single-species VA system and Vlasov-Maxwell (VM) system.
The main feature of the schemes is their ability to preserve the total particle
number and total energy on the fully discrete level regardless of mesh size.
Those are desired properties of numerical schemes especially for long time
simulations with under-resolved mesh. The conservation is realized by explicit
and implicit energy-conserving temporal discretizations, and the discontinuous
Galerkin (DG) spatial discretizations. We benchmarked our algorithms on a test
example to check the one-species limit, and the current-driven ion-acoustic
instability. To simulate the current-driven ion-acoustic instability, a slight
modification for the implicit method is necessary to fully decouple the split
equations. This is achieved by a Gauss-Seidel type iteration technique.
Numerical results verified the conservation and performance of our methods
The Energy Conserving Particle-in-Cell Method
A new Particle-in-Cell (PIC) method, that conserves energy exactly, is
presented. The particle equations of motion and the Maxwell's equations are
differenced implicitly in time by the midpoint rule and solved concurrently by
a Jacobian-free Newton Krylov (JFNK) solver. Several tests show that the finite
grid instability is eliminated in energy conserving PIC simulations, and the
method correctly describes the two-stream and Weibel instabilities, conserving
exactly the total energy. The computational time of the energy conserving PIC
method increases linearly with the number of particles, and it is rather
insensitive to the number of grid points and time step. The kinetic enslavement
technique can be effectively used to reduce the problem matrix size and the
number of JFNK solver iterations
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