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stability of alternate direction implicit scheme

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Old   September 21, 2012, 05:41
Default stability of alternate direction implicit scheme
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Aniket Sachdeva
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In the ADI scheme for solving 2 dimensional unsteady diffusion problem, the pentadiagonal matrix is expressed as sum of two tridiagonal matrices along horizontal and vertical directions at half and full time steps respectively. For stability of a numerical scheme all the contribution coefficients must be positive. But, in this case the coefficient of Tn (Temperature at nth time level) in the expression of Tn+1 has a coefficient 1-gamma*dt/dx^2, where gamma is the diffusion coefficient, dt full time step value. This can become negative for gamma > dx^2/dt, thus imposing a stability criterion in the implicit scheme. Is this true? Can implicit schemes also be unstable in some cases, or am I making some mistake?
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