Five Ways of Reducing the Crank-Nicolson Oscillations

Østerby, Ole

doi:10.7146/dpb.v31i558.7115

Cited by 13 publications

(16 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Although we have not observed this problematic behavior when solving scalar ODEs, ringing effects are often observed for very stiff PDEs (typically with very small spatial grid sizes to resolve fine detail) with relatively large tolerances on the time step or towards the end of a simulation when close to steady state. Situations such as these are discussed by Osterby [22] along with a variety of means of suppressing the oscillations. Our code implements an alternative strategy-time step averaging.…”

Section: P Gresho D Griffiths and D Silvestermentioning

confidence: 99%

Adaptive Time-Stepping for Incompressible Flow Part I: Scalar Advection-Diffusion

Gresho¹,

Griffiths²,

Silvester³

2008

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. Even the simplest advection-diffusion problems can exhibit multiple time scales. This means that robust variable step time integrators are a prerequisite if such problems are to be efficiently solved computationally. The performance of the second order trapezoid rule using an explicit Adams-Bashforth method for error control is assessed in this work. This combination is particularly well suited to long time integration of advection-dominated problems. Herein it is shown that a stabilized implementation of the trapezoid rule leads to a very effective integrator in other situations: specifically diffusion problems with rough initial data; and general advection-diffusion problems with different physical time scales governing the system evolution.Key words. time-stepping, adaptivity, convection-diffusion AMS subject classifications. 65M12, 65M15, 65M20 DOI. 10.1137/0706880181. Background and context. The adaptive time-stepping algorithm that is the focus of this work is certainly not new. We consider the simplest Adams-BashforthMoulton pair. A version of our algorithm is hard-wired as the MATLAB function ode23t, see [24], and the underlying methodology is discussed in any many textbooks on the numerical solution of ODEs. See, for example, Henrici [13, p. 258] where estimation of the truncation error is discussed, or Iserles [16, p. 78], where stepdoubling and halving is described.The aim of this work is to assess the performance of this integrator in the context of method-of-lines discretization of PDEs that arise in incompressible flow modelling. In particular, we hope to provide insight into the role of adaptive time-stepping in the context of modelling multiple physical timescales. For this purpose it suffices to restrict our attention to the following simple model of scalar advection-diffusion:together with the initial condition u(x, 0) = u 0 (x), and boundary conditions (BCs)where a ≥ 0 (the advecting velocity), ν ≥ 0 (diffusivity), and u L and u R are given constants. In part II, we build on the foundation laid in this paper and consider the potential of the integrator in the context of solving the Navier-Stokes equations. *

show abstract

Section: P Gresho D Griffiths and D Silvestermentioning

confidence: 99%

Adaptive Time-Stepping for Incompressible Flow Part I: Scalar Advection-Diffusion

Gresho¹,

Griffiths²,

Silvester³

2008

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…This is not an instability due to an error propagation as one might get using an unstable simulation method; this is shown by the points on two of the curves, obtained by the eigenvalue-eigenvector method, which computes a current directly for each T value, without stepping with time intervals. This is the reason that the word "waves" is used here, rather than "oscillation", such as one obtains with the Crank-Nicolson method [79] applied to problems with an initial transient [80,81], which are indeed oscillatory error propagation effects. We have no explanation for this puzzling effect, which is clearly a property of the transformed grid.…”

Section: Verbrugge-baker Transformation Vbmentioning

confidence: 99%

Minimum grid digital simulation of chronoamperometry at a disk electrode

Britz

Østerby

Strutwolf

2012

Electrochimica Acta

Self Cite

View full text Add to dashboard Cite

a b s t r a c tWe seek to find the best ways to achieve a minimum grid with a given target accuracy in the computed current in the simulation of a diffusion limited chronoamperometric experiment at an ultramicrodisk electrode. We present some results for direct discretisation on the cylindrical geometry in (R, Z) space and in transformed coordinates (Â, ), comparing four commonly used transformations. In all cases, the use of multi-point spatial derivative approximations are explored to find an optimum. Orthogonal collocation is studied in the two dimensions, and found less efficient than an evenly divided grid and smaller multi-point approximations. The eigenvalue-eigenvector method is studied and found to be relatively inefficient but was useful in providing some information on error waves seen with three of the four transformations used. These error waves are not due to an error propagation, a fact that became clear from the eigenvalue-eigenvector method, which reproduced the waves.

show abstract

“…Ringing-induced stall is typical in an unsteady system approaching steady state and often occurs following abrupt temporal changes (as by forcing terms). The ringing instability may worsen for stiffer problems [37,42,52] as well as problems where space-time coupling is stronger.…”

Section: Review Of Trmentioning

confidence: 99%

“…The nature of this ringing and strategies for mitigating its effects have been discussed extensively in the literature [40,46,45,58,27,20,10], especially in the context of the second order finite difference method-of-lines (the well-known CrankNicolson method [19]) [11,41,42], and more recently of the finite element method-oflines [33,2,50,37,24,25,52]. For our purpose, the most relevant method is the time step averaging (TSA) method due to Gresho, Griffiths, and Silvester [31], a simple and inexpensive scheme that effectively stabilizes the solution, albeit with compromised accuracy.…”

mentioning

confidence: 99%

A New Stabilization of Adaptive Step Trapezoid Rule Based on Finite Difference Interrupts

Lee

Nam²,

Pasquali³

2015

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. The adaptive step trapezoid rule (TR) is a generally effective numerical integrator, but it is prone to ringing instability and solution stall. We introduce a new stabilization algorithm based on finite difference interrupts (FDI) that reduces ringing and prevents stall. Unlike previously reported stabilization schemes, our algorithm achieves stability and second order accuracy without incurring significant computational cost or spurious diffusion. TR with FDI is at least as stable and accurate as existing methods when solving the prototypical scalar problem. We demonstrate that it has better stability, accuracy, and cost when solving the tightly coupled system describing free surface flows of viscoelastic liquids. Though we demonstrate TR with FDI in the context of a finite element method-of-lines, it is applicable to any TR-based algorithm.

show abstract

Five Ways of Reducing the Crank-Nicolson Oscillations

Cited by 13 publications

References 3 publications

Adaptive Time-Stepping for Incompressible Flow Part I: Scalar Advection-Diffusion

Adaptive Time-Stepping for Incompressible Flow Part I: Scalar Advection-Diffusion

Minimum grid digital simulation of chronoamperometry at a disk electrode

A New Stabilization of Adaptive Step Trapezoid Rule Based on Finite Difference Interrupts

Contact Info

Product

Resources

About