# Factorized Runge-Kutta-Chebyshev Methods

@article{OSullivan2017FactorizedRM, title={Factorized Runge-Kutta-Chebyshev Methods}, author={Stephen O'Sullivan}, journal={arXiv: Numerical Analysis}, year={2017} }

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

#### Paper Mentions

#### 2 Citations

Runge-Kutta-Gegenbauer explicit methods for advection-diffusion problems

- Mathematics, Physics
- J. Comput. Phys.
- 2019

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

- Computer Science, Mathematics
- J. Comput. Phys.
- 2019

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

- Mathematics, Computer Science
- J. Comput. Phys.
- 2015

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

- Mathematics, Computer Science
- Comput. Phys. Commun.
- 2009

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

- Mathematics, Computer Science
- Numerische Mathematik
- 2001

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

- Mathematics, Computer Science
- J. Comput. Phys.
- 2014

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

- Mathematics
- 1996

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, at… Expand

RKC: an explicit solver for parabolic PDEs

- Mathematics
- 1997

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 formulas… Expand

Optimized high-order splitting methods for some classes of parabolic equations

- Computer Science, Mathematics
- Math. Comput.
- 2013

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

- Physics
- 2012

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

- Mathematics
- 1996

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 which… Expand

On the Internal Stability of Explicit, m‐Stage Runge‐Kutta Methods for Large m‐Values

- Mathematics
- 1979

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 parabolic… Expand