554 research outputs found
An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations
An implicit-explicit (IMEX) extension of the explicit Runge-Kutta-Chebyshev (RKC) scheme designed for parabolic PDEs is proposed for diffusion-reaction problems with severely stiff reaction terms. The IMEX scheme treats these reaction terms implicitly and diffusion terms explicitly. Within the setting of linear stability theory, the new IMEX scheme is unconditionally stable for reaction terms having a Jacobian matrix with a real spectrum. For diffusion terms the stability characteristics remain unchanged. A numerical comparison for a stiff, nonlinear radiation-diffusion problem between an RKC solver, an IMEX-RKC solver and the popular implicit BDF solver VODPK using the Krylov solver GMRES illustrates the excellent performance of the new scheme
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 is presented. Roots of FRKC
stability polynomials of degree are used to construct explicit schemes
comprising forward Euler stages with internal stability ensured through a
sequencing algorithm which limits the internal amplification factors to . The associated stability domain scales as 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
Spatially partitioned embedded Runge-Kutta Methods
We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory
Runge-Kutta-Gegenbauer explicit methods for advection-diffusion problems
In this paper, Runge-Kutta-Gegenbauer (RKG) stability polynomials of
arbitrarily high order of accuracy are introduced in closed form. The stability
domain of RKG polynomials extends in the the real direction with the square of
polynomial degree, and in the imaginary direction as an increasing function of
Gegenbauer parameter. Consequently, the polynomials are naturally suited to the
construction of high order stabilized Runge-Kutta (SRK) explicit methods for
systems of PDEs of mixed hyperbolic-parabolic type.
We present SRK methods composed of ordered forward Euler stages, with
complex-valued stepsizes derived from the roots of RKG stability polynomials of
degree . Internal stability is maintained at large stage number through an
ordering algorithm which limits internal amplification factors to .
Test results for mildly stiff nonlinear advection-diffusion-reaction problems
with moderate () mesh P\'eclet numbers are provided at second,
fourth, and sixth orders, with nonlinear reaction terms treated by complex
splitting techniques above second order.Comment: 20 pages, 7 figures, 3 table
On stabilized integration for time-dependent PDEs
An integration method is discussed which has been designed totreat parabolic and hyperbolic terms explicitly and stiff reaction terms implicitly. The method is a special
two-step form of the one-step IMEX (Implicit-Explicit) RKC (Runge-Kutta-Chebyshev) method. The special two-step form is introduced with the aim of getting a non-zero
imaginary stability boundary which is zero for the one-step method. Having a non-zero imaginary stability boundary allows, for example, the integration of pure advection
equations space-discretized with centered schemes, the integration of damped or viscous wave equations, the integration of coupled sound and heat flow equations, etc.
For our class of methods it also simplifies the choice of temporal step sizes satisfying the Von Neumann stability criterion, by embedding a thin long rectangle inside the
stability region. Embedding rectangles or other tractable domains with this purpose is an idea of Wesselin
Fourth-order time-stepping for stiff PDEs on the sphere
We present in this paper algorithms for solving stiff PDEs on the unit sphere
with spectral accuracy in space and fourth-order accuracy in time. These are
based on a variant of the double Fourier sphere method in coefficient space
with multiplication matrices that differ from the usual ones, and
implicit-explicit time-stepping schemes. Operating in coefficient space with
these new matrices allows one to use a sparse direct solver, avoids the
coordinate singularity and maintains smoothness at the poles, while
implicit-explicit schemes circumvent severe restrictions on the time-steps due
to stiffness. A comparison is made against exponential integrators and it is
found that implicit-explicit schemes perform best. Implementations in MATLAB
and Chebfun make it possible to compute the solution of many PDEs to high
accuracy in a very convenient fashion
Solving reaction-diffusion equations 10 times faster
The most popular numerical method for solving systems of reaction-diffusion equations continues to be a low order finite-difference scheme coupled with low order Euler time stepping. This paper extends previous 1D work and reports experiments that show that with high--order methods one can speed up such simulations for 2D and 3D problems by factors of 10--100. A short MATLAB code (2/3D) that can serve as a template is included.\ud
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This work was supported by the Engineering and Physical Sciences Research Council (UK) and by the MathWorks, Inc
Explicit stabilized multirate method for stiff differential equations
Stabilized Runge–Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized Runge–Kutta methods overcome the stringent stability condition of standard methods without sacrificing explicitness. However, when stiffness is only induced by a few components, as in the presence of spatially local mesh refinement, their efficiency deteriorates. To remove the crippling effect of a few severely stiff components on the entire system of differential equations, we derive a modified equation, whose stiffness solely depends on the remaining mildly stiff components. By applying stabilized Runge–Kutta methods to this modified equation, we then devise an explicit multirate Runge–Kutta–Chebyshev (mRKC) method whose stability conditions are independent of a few severely stiff components. Stability of the mRKC method is proved for a model problem, whereas its efficiency and usefulness are demonstrated through a series of numerical experiments
Accelerating moderately stiff chemical kinetics in reactive-flow simulations using GPUs
The chemical kinetics ODEs arising from operator-split reactive-flow
simulations were solved on GPUs using explicit integration algorithms. Nonstiff
chemical kinetics of a hydrogen oxidation mechanism (9 species and 38
irreversible reactions) were computed using the explicit fifth-order
Runge-Kutta-Cash-Karp method, and the GPU-accelerated version performed faster
than single- and six-core CPU versions by factors of 126 and 25, respectively,
for 524,288 ODEs. Moderately stiff kinetics, represented with mechanisms for
hydrogen/carbon-monoxide (13 species and 54 irreversible reactions) and methane
(53 species and 634 irreversible reactions) oxidation, were computed using the
stabilized explicit second-order Runge-Kutta-Chebyshev (RKC) algorithm. The
GPU-based RKC implementation demonstrated an increase in performance of nearly
59 and 10 times, for problem sizes consisting of 262,144 ODEs and larger, than
the single- and six-core CPU-based RKC algorithms using the
hydrogen/carbon-monoxide mechanism. With the methane mechanism, RKC-GPU
performed more than 65 and 11 times faster, for problem sizes consisting of
131,072 ODEs and larger, than the single- and six-core RKC-CPU versions, and up
to 57 times faster than the six-core CPU-based implicit VODE algorithm on
65,536 ODEs. In the presence of more severe stiffness, such as ethylene
oxidation (111 species and 1566 irreversible reactions), RKC-GPU performed more
than 17 times faster than RKC-CPU on six cores for 32,768 ODEs and larger, and
at best 4.5 times faster than VODE on six CPU cores for 65,536 ODEs. With a
larger time step size, RKC-GPU performed at best 2.5 times slower than six-core
VODE for 8192 ODEs and larger. Therefore, the need for developing new
strategies for integrating stiff chemistry on GPUs was discussed.Comment: 27 pages, LaTeX; corrected typos in Appendix equations A.10 and A.1
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