An alternative TLM method for steady‐state convection–diffusion

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SUMMARYThis paper describes how the lossy transmission line modelling (TLM) method for diffusion can be extended to solve the convection-diffusion equation. The method is based on the correspondence between the convection-diffusion equation and the equation for the voltage on a lossy transmission line with properties varying exponentially over space. It is unconditionally stable and converges rapidly to highly accurate steady-state solutions for a wide range of Peclet numbers from low to high. The method solves the non-conservative form of the convection-diffusion equation but it is shown how it can be modified to solve the conservative form. Under transient conditions the TLM scheme exhibits significant numerical diffusion and numerical convection leading to poor accuracy, but both these errors go to zero as a solution approaches steady state.

“…It has been found analytically [16] that under purely transient conditions (i.e. for convectiondiffusion in an infinite medium) the relationships are…”

Section: Transient Resultsmentioning

confidence: 98%

“…The voltage on such a TL can be modelled using the TLM method, thus providing a numerical solution for Equation (16).…”

Section: The Methodsmentioning

confidence: 99%

A transmission line modelling (TLM) method for steady‐state convection–diffusion

Kennedy

O’Connor

2007

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“…There is no reason why it cannot be similarly applied to more complex TL networks [11]. It can be used for parameterization and can also yield equations for node voltage distribution variance errors.…”

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

“…What is not clear is whether this improvement is sustained over longer numbers of time steps. In order to answer this question, the models were run for 1000 time steps and the maximum absolute relative error in k Vn was calculated at each time step using the formula [11] where k Vn AN AL n is the analytical solution at node n and time step k. This error value is plotted against k for = 0.35 in Figure 8 for an LR model. It is clear that adjusting 2 gives improved results at every time step, with the sole exception of time step 2.…”

Section: Infinite Medium With Gaussian-shaped Initial Concentration Pmentioning

confidence: 99%

See 1 more Smart Citation

Error analysis and reduction in lossy TLM

Kennedy¹,

O’Connor²

2007

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A TLM method for steady‐state convection–diffusion: Some additions and refinements

Kennedy

O’Connor

2008

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SUMMARYA recent paper introduced a novel and efficient scheme, based on the transmission line modelling (TLM) method, for solving steady-state convection-diffusion problems. This paper shows how this onedimensional scheme can be adapted to include reaction and source terms and how it can be implemented with non-equidistant nodes. It introduces new ways of calculating the necessary model parameters which can improve the accuracy of the scheme, shows how steady-state solutions can be obtained directly, and compares results with those from two finite difference (FD) methods. While the cost of implementation is higher than for the FD schemes, the new TLM scheme can be significantly more accurate, especially when convection dominates.