28,030 research outputs found
Towards optimal explicit time-stepping schemes for the gyrokinetic equations
The nonlinear gyrokinetic equations describe plasma turbulence in laboratory
and astrophysical plasmas. To solve these equations, massively parallel codes
have been developed and run on present-day supercomputers. This paper describes
measures to improve the efficiency of such computations, thereby making them
more realistic. Explicit Runge-Kutta schemes are considered to be well suited
for time-stepping. Although the numerical algorithms are often highly
optimized, performance can still be improved by a suitable choice of the
time-stepping scheme, based on spectral analysis of the underlying operator.
Here, an operator splitting technique is introduced to combine first-order
Runge-Kutta-Chebychev schemes for the collision term with fourth-order schemes
for the remaining terms. In the nonlinear regime, based on the observation of
eigenvalue shifts due to the (generalized) advection term, an
accurate and robust estimate for the nonlinear timestep is developed. The
presented techniques can reduce simulation times by factors of up to three in
realistic cases. This substantial speedup encourages the use of similar
timestep optimized explicit schemes not only for the gyrokinetic equation, but
also for other applications with comparable properties.Comment: 11 pages, 5 figures, accepted for publication in Computer Physics
Communication
Advanced operator-splitting-based semi-implicit spectral method to solve the binary phase-field crystal equations with variable coefficients
We present an efficient method to solve numerically the equations of dissipative dynamics of the binary phase-field crystal model proposed by Elder et al. [Phys. Rev. B 75, 064107 (2007)] characterized by variable coefficients. Using the operator splitting method, the problem has been decomposed into sub-problems that can be solved more efficiently. A combination of non-trivial splitting with spectral semi-implicit solution leads to sets of algebraic equations of diagonal matrix form. Extensive testing of the method has been carried out to find the optimum balance among errors associated with time integration, spatial discretization, and splitting. We show that our method speeds up the computations by orders of magnitude relative to the conventional explicit finite difference scheme, while the costs of the pointwise implicit solution per timestep remains low. Also we show that due to its numerical dissipation, finite differencing can not compete with spectral differencing in terms of accuracy. In addition, we demonstrate that our method can efficiently be parallelized for distributed memory systems, where an excellent scalability with the number of CPUs is observed
High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation
We construct a high order discontinuous Galerkin method for solving general
hyperbolic systems of conservation laws. The method is CFL-less, matrix-free,
has the complexity of an explicit scheme and can be of arbitrary order in space
and time. The construction is based on: (a) the representation of the system of
conservation laws by a kinetic vectorial representation with a stiff relaxation
term; (b) a matrix-free, CFL-less implicit discontinuous Galerkin transport
solver; and (c) a stiffly accurate composition method for time integration. The
method is validated on several one-dimensional test cases. It is then applied
on two-dimensional and three-dimensional test cases: flow past a cylinder,
magnetohydrodynamics and multifluid sedimentation
Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution
We present a new solver for nonlinear parabolic problems that is L-stable and
achieves high order accuracy in space and time. The solver is built by first
constructing a single-dimensional heat equation solver that uses fast O(N)
convolution. This fundamental solver has arbitrary order of accuracy in space,
and is based on the use of the Green's function to invert a modified Helmholtz
equation. Higher orders of accuracy in time are then constructed through a
novel technique known as successive convolution (or resolvent expansions).
These resolvent expansions facilitate our proofs of stability and convergence,
and permit us to construct schemes that have provable stiff decay. The
multi-dimensional solver is built by repeated application of dimensionally
split independent fundamental solvers. Finally, we solve nonlinear parabolic
problems by using the integrating factor method, where we apply the basic
scheme to invert linear terms (that look like a heat equation), and make use of
Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our
solver is applied to several linear and nonlinear equations including heat,
Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two
dimensions
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