This paper explores the utility of a sixth-order compact finite difference (CFD6) scheme for the solution of the sine-Gordon equation. The CFD6 scheme in space and a third-order strong stability preserving Runge-Kutta scheme in time have been combined for solving the equation. This scheme needs less storage space, as opposed to the conventional numerical methods, and causes to less accumulation of numerical errors. The scheme is implemented to solve three test problems having exact solutions. Comparisons of the computed results with exact solutions showed that the method is capable of achieving high accuracy with minimal computational effort. The present results are also seen to be more accurate than some available results given in the literature. The scheme is seen to be a very reliable alternative technique to existing ones. Copyright © 2009 John Wiley & Sons, Ltd.
jueves, 5 de noviembre de 2009
A sixth-order compact finite difference method for the one-dimensional sine-Gordon equation
This paper explores the utility of a sixth-order compact finite difference (CFD6) scheme for the solution of the sine-Gordon equation. The CFD6 scheme in space and a third-order strong stability preserving Runge-Kutta scheme in time have been combined for solving the equation. This scheme needs less storage space, as opposed to the conventional numerical methods, and causes to less accumulation of numerical errors. The scheme is implemented to solve three test problems having exact solutions. Comparisons of the computed results with exact solutions showed that the method is capable of achieving high accuracy with minimal computational effort. The present results are also seen to be more accurate than some available results given in the literature. The scheme is seen to be a very reliable alternative technique to existing ones. Copyright © 2009 John Wiley & Sons, Ltd.
Non-local implicit gradient damage models with low-order finite elements
Driven by non-local strains, which are approximated at least by a linear function, the damage evolution continuously varies, whereas the distribution of local strains remains constant, and this mismatch leads to stress oscillations within a low-order finite element. Through the enforcement of constant damage via an appropriate average of the non-local strains that drive damage, the latter becomes discontinuous across inter-elements. The consistency of the finite element formulation can be retrieved, and it hence enables a significant reduction of the occurrence of stress oscillations, whereas the convergence rate of the finite element formulation is still preserved. Numerical examples with 1D and 2D typical examples show the efficiency and the performance of the proposed approach. Copyright © 2009 John Wiley & Sons, Ltd.
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