Article

# The method of false transient for the solution of coupled elliptic equations

School of Mechanical and Industrial Engineering, University of New South Wales, Kensington, Australia 2033
(Impact Factor: 2.43). 08/1973; 12(4):435-461. DOI: 10.1016/0021-9991(73)90097-1

ABSTRACT A method for the numerical solution of a system of coupled, nonlinear elliptic partial differential equations is described, and the application of the method to the equations governing steady, laminar natural convection is presented. The essential feature of the method is the conversion of the equations to a parabolic form by the addition of false time derivatives, thus, enabling a marching solution, equivalent to a single iterative procedure, to be used. The method is evaluated by applying it to a well known two-dimensional problem and some examples of its use in three dimensions are given.

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• "To solve the system of partial differential equations in the above section, we have employed a false transient method (Mallinson and de Vahl Davis, 1973) by which the nonlinear equations are transformed to parabolic form, and the transformed equations are then discretized on a non-uniform grid. The resulting finite difference equations have been solved by a well-known alternating direction implicit (ADI) method, following Singh et al. (2000). "
##### Dataset: A numerical investigation of a buoyancy driven flow in a semi-porous cavity: Comparative effects of ramped and isothermal wall conditions
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ABSTRACT: Steady two-dimensional natural convection taking place in a rectangular cavity, partially filled with an iso-tropic porous material, has been investigated numerically using an ADI method. It is assumed that one of the vertical walls of the cavity has a ramped temperature distribution. The vorticity-stream function formulation has been used to solve the set of nonlinear partial differential equations governing the flows in the clear region and the adjoining porous region. The effects of Darcy number and Rayleigh number have been discussed in detail.
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• "In this study, the method of modified dynamics or false transients (e.g. [1] [2]) is applied to obtain the structure of a steady flow. The governing equations (1) and (2) are modified as "
##### Article: ADI method based on C2-continuous two-node integrated-RBF elements for viscous flows
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ABSTRACT: We propose a C2C2-continuous alternating direction implicit (ADI) method for the solution of the streamfunctionâ€“vorticity equations governing steady 2D incompressible viscous fluid flows. Discretisation is simply achieved with Cartesian grids. Local two-node integrated radial basis function elements (IRBFEs) [D.-A. An-Vo, N. Mai-Duy, T. Tran-Cong, A C2C2-continuous control-volume technique based on Cartesian grids and two-node integrated-RBF elements for second-order elliptic problems, CMES: Comput. Model. Eng. Sci. 72 (2011) 299â€“334] are used for the discretisation of the diffusion terms, and then the convection terms are incorporated into system matrices by treating nodal derivatives as unknowns. ADI procedure is applied for the time integration. Following ADI factorisation, the two-dimensional problem becomes a sequence of one-dimensional problems. The solution strategy consists of multiple use of a one-dimensional sparse matrix algorithm that helps saving the computational cost. High levels of accuracy and efficiency of the present methods are demonstrated with solutions of several benchmark problems defined on rectangular and non-rectangular domains.
Applied Mathematical Modelling 04/2013; 37(7):5184-5203. DOI:10.1016/j.apm.2012.10.034 · 2.25 Impact Factor
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• "To solve the system of partial differential equations in the above section, we have employed a false transient method (Mallinson and de Vahl Davis, 1973) by which the nonlinear equations are transformed to parabolic form, and the transformed equations are then discretized on a non-uniform grid. The resulting finite difference equations have been solved by a well-known alternating direction implicit (ADI) method, following Singh et al. (2000). "
##### Article: Chandran, P., Sacheti, N. C. and Singh, A. K., A numerical investigation of a buoyancy driven flow in a semi-porous cavity: Comparative effects of ramped and isothermal wall conditions, Journal of Hydrology and Hydromechanics, Vol. 61(2013), pp. 103-111.
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ABSTRACT: Steady two-dimensional natural convection taking place in a rectangular cavity, partially filled with an isotropic porous material, has been investigated numerically using an ADI method. It is assumed that one of the vertical walls of the cavity has a ramped temperature distribution. The vorticity-stream function formulation has been used to solve the set of nonlinear partial differential equations governing the flows in the clear region and the adjoining porous region. The effects of Darcy number and Rayleigh number have been discussed in detail.
Journal of Hydrology and Hydromechanics 01/2013; · 1.49 Impact Factor