A class of high-order Runge-Kutta-Chebyshev stability polynomials

The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC) stability polynomials of arbitrary order $N$ is presented. Roots of FRKC stability polynomials of degree $L=MN$ are used to construct explicit schemes comprising $L$ forward Euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to $\sim L^2$. The associated stability domain scales as $M^2$ along the real axis. Marginally stable real-valued points on the interior of the stability domain are removed via a prescribed damping procedure. By construction, FRKC schemes meet all linear order conditions; for nonlinear problems at orders above 2, complex splitting or Butcher series composition methods are required. Linear order conditions of the FRKC stability polynomials are verified at orders 2, 4, and 6 in numerical experiments. Comparative studies with existing methods show the second-order unsplit FRKC2 scheme and higher order (4 and 6) split FRKCs schemes are efficient for large moderately stiff problems.Comment: 24 pages, 5 figures. Accepted for publication in Journal of Computational Physics, 22 Jul 2015. Revise

High order time integrators for the simulation of charged particle motion in magnetic quadrupoles

Magnetic quadrupoles are essential components of particle accelerators like the Large Hadron Collider. In order to study numerically the stability of the particle beam crossing a quadrupole, a large number of particle revolutions in the accelerator must be simulated, thus leading to the necessity to preserve numerically invariants of motion over a long time interval and to a substantial computational cost, mostly related to the repeated evaluation of the magnetic vector potential. In this paper, in order to reduce this cost, we first consider a specific gauge transformation that allows to reduce significantly the number of vector potential evaluations. We then analyze the sensitivity of the numerical solution to the interpolation procedure required to compute magnetic vector potential data from gridded precomputed values at the locations required by high order time integration methods. Finally, we compare several high order integration techniques, in order to assess their accuracy and efficiency for these long term simulations. Explicit high order Lie methods are considered, along with implicit high order symplectic integrators and conventional explicit Runge Kutta methods. Among symplectic methods, high order Lie integrators yield optimal results in terms of cost/accuracy ratios, but non symplectic Runge Kutta methods perform remarkably well even in very long term simulations. Furthermore, the accuracy of the field reconstruction and interpolation techniques are shown to be limiting factors for the accuracy of the particle tracking procedures.Comment: 39 pages, 18 figure

More efficient time integration for Fourier pseudo-spectral DNS of incompressible turbulence

Time integration of Fourier pseudo-spectral DNS is usually performed using the classical fourth-order accurate Runge--Kutta method, or other methods of second or third order, with a fixed step size. We investigate the use of higher-order Runge-Kutta pairs and automatic step size control based on local error estimation. We find that the fifth-order accurate Runge--Kutta pair of Bogacki \& Shampine gives much greater accuracy at a significantly reduced computational cost. Specifically, we demonstrate speedups of 2x-10x for the same accuracy. Numerical tests (including the Taylor-Green vortex, Rayleigh-Taylor instability, and homogeneous isotropic turbulence) confirm the reliability and efficiency of the method. We also show that adaptive time stepping provides a significant computational advantage for some problems (like the development of a Rayleigh-Taylor instability) without compromising accuracy

Finite volume methods for unidirectional dispersive wave models

We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular we consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves and their various interactions.Comment: 25 pages, 12 figures, 51 references. Other authors papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

Finite volume methods for unidirectional dispersive wave model

We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular, we consider a KdV–BBM-type equation. Explicit and implicit–explicit Runge–Kutta-type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants’ conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves, and their various interaction

Computation of saddle type slow manifolds using iterative methods

This paper presents an alternative approach for the computation of trajectory segments on slow manifolds of saddle type. This approach is based on iterative methods rather than collocation-type methods. Compared to collocation methods, that require mesh refinements to ensure uniform convergence with respect to $\epsilon$, appropriate estimates are directly attainable using the method of this paper. The method is applied to several examples including: A model for a pair of neurons coupled by reciprocal inhibition with two slow and two fast variables and to the computation of homoclinic connections in the FitzHugh-Nagumo system.Comment: To appear in SIAM Journal of Applied Dynamical System