The Extrapolation of First Order Methods for Parabolic Partial Differential Equations, II

Gourlay, A. R.; Morris, J.

doi:10.1137/0717054

Cited by 90 publications

(23 citation statements)

References 4 publications

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“…All computations were run using IEEE 754 Standard double precision, giving approximately 16-decimal precision. The time marching simulations were done using, for the time integration, three-point backward differentiation formula (BDF) [68][69][70] started with one backwards implicit or Laasonen [65] (BI) step; and by BI with extrapolation [66,67,75]; as well as by the Douglas-Rachford method [72,73]. All three methods produced very similar results and those for BI/BDF are those presented in the tables.…”

Section: Computers and Methodsmentioning

confidence: 77%

Minimum grid digital simulation of chronoamperometry at a disk electrode

Britz

Østerby

Strutwolf

2012

Electrochimica Acta

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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.

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Section: Computers and Methodsmentioning

confidence: 77%

Minimum grid digital simulation of chronoamperometry at a disk electrode

Britz

Østerby

Strutwolf

2012

Electrochimica Acta

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“…In the case of expanding time intervals (see later), this was not convenient, as unequal time intervals reduce the order of three-point BDF to O(ıT), so four-point BDF would be needed to achieve the same order [38]. It was more convenient in this case to use backward implicit with extrapolation [39][40][41][42], which is also O(ıT 2 ) (for the simple second-order extrapolation method). The method requires three operations per time step rather than one, but as comparatively very few time steps were needed if they were expanding with time, computation was extremely fast, as will be seen.…”

Section: Discretisation Simulation Methodsmentioning

confidence: 99%

Several ways to simulate time dependent liquid junction potentials by finite differences

Britz

Strutwolf

2014

Electrochimica Acta

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a b s t r a c tSeveral ways of simulating time-dependent migration effects at an electrolytic liquid junction are explored, for a simple uni-univalent electrolyte, both with the full Nernst-Planck-Poisson (NPP) equation set and the reduced set from the electroneutrality condition (ENC) assumption. Using the NPP approach, the system can be simulated using all three variables (method AB), the two concentrations and the potential, or the two concentrations and the potential field (method ABE), or in principle by substituting for the potential field, thereby reducing to the two concentration variables (method AB). The two first methods are about equally efficient, whereas the latter method is seen to be quite inaccurate. Results at long times compare very well with the Henderson equation. Using the full NPP set, junction potentials are time-dependent but not when applying the ENC, where the potential rises to the Henderson value immediately. Results for KCl and HCl are presented, with left/right concentration ratios equal to 0.1 in both cases.

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“…Thus the boundary conditions u(0, t) and u(X, t) can be determined using (2) and (3) with (6) respectively. Applying (1) to all the interior mesh points within R at time level t = nl with the space derivative replaced by (5) leads to a system of N first-order ODE's of the form…”

Section: Derivation Of the Schemementioning

confidence: 99%