Factorized Runge-Kutta-Chebyshev Methods

@article{OSullivan2017FactorizedRM,
  title={Factorized Runge-Kutta-Chebyshev Methods},
  author={Stephen O'Sullivan},
  journal={arXiv: Numerical Analysis},
  year={2017}
}
  • S. O'Sullivan
  • Published 2017
  • Computer Science, Mathematics, Physics
  • arXiv: Numerical Analysis
The second-order extended stability Factorized Runge-Kutta-Chebyshev (FRKC2) class of explicit schemes for the integration of large systems of PDEs with diffusive terms is presented. FRKC2 schemes are straightforward to implement through ordered sequences of forward Euler steps with complex stepsizes, and easily parallelised for large scale problems on distributed architectures. Preserving 7 digits for accuracy at 16 digit precision, the schemes are theoretically capable of maintaining… Expand

Figures and Tables from this paper

Paper Mentions

Runge-Kutta-Gegenbauer explicit methods for advection-diffusion problems
TLDR
SRK methods are presented, composed of L ordered forward Euler stages, with complex-valued stepsizes derived from the roots of RKG stability polynomials of degree L, for systems of PDEs of mixed hyperbolic-parabolic type. Expand
Paired explicit Runge-Kutta schemes for stiff systems of equations
  • B. Vermeire
  • Computer Science, Mathematics
  • J. Comput. Phys.
  • 2019
TLDR
It is demonstrated that P-ERK schemes can significantly accelerate the solution of stiff systems of equations when using an explicit approach, and that they maintain accuracy with respect to conventional Runge-Kutta methods and available reference data. Expand

References

SHOWING 1-10 OF 35 REFERENCES
A class of high-order Runge-Kutta-Chebyshev stability polynomials
TLDR
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 and complex splitting or Butcher series composition methods are required. Expand
Second-order stabilized explicit Runge-Kutta methods for stiff problems
TLDR
This paper shows that stabilized Runge–Kutta methods have some difficulties to solve efficiently problems where the eigenvalues are very large in absolute value, and derives second-order methods with up to 320 stages and good stability properties. Expand
Second order Chebyshev methods based on orthogonal polynomials
TLDR
The aim of this paper is to show that with the use of orthogonal polynomials, the authors can construct nearly optimal stability polynmials of second order with a three-term recurrence relation. Expand
A stabilized Runge-Kutta-Legendre method for explicit super-time-stepping of parabolic and mixed equations
TLDR
This work builds temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Legendre polynomials and proves that the newly-designed RKL1 and RKl2 schemes have a very desirable monotonicity preserving property for one-dimensional problems - a solution that is monotone at the beginning of a time step retains that property at the end of that time step. Expand
Super-time-stepping acceleration of explicit schemes for parabolic problems
The goal of the paper is to bring to the attention of the computational community a long overlooked, very simple, acceleration method that impressively speeds up explicit time-stepping schemes, atExpand
RKC: an explicit solver for parabolic PDEs
The FORTRAN program RKC is intended for the time integration of parabolic partial differential equations discretized by the method of lines. It is based on a family of Runge-Kutta-Chebyshev formulasExpand
Optimized high-order splitting methods for some classes of parabolic equations
TLDR
It is shown that, in the general case, 14 is not an order barrier for splitting method s with complex coefficients with positive real part by building explicitly a method of order 16 as a composition of methods of order 8. Expand
A second-order accurate Super TimeStepping formulation for anisotropic thermal conduction
Astrophysical fluid dynamical problems rely on efficient numerical solution techniques for hyperbolic and parabolic terms. Efficient techniques are available for treating the hyperbolic terms.Expand
Explicit Runge-Kutta methods for parabolic partial differential equations
Abstract Numerical methods for parabolic PDEs have been studied for many years. A great deal of the research focuses on the stability problem in the time integration of the systems of ODEs whichExpand
On the Internal Stability of Explicit, m‐Stage Runge‐Kutta Methods for Large m‐Values
Explicit, m-stage Runge-Kutta methods are derived for which the maximal stable integration step per right hand side evaluation is proportional to m when applied to semi-discrete parabolicExpand
...
1
2
3
4
...