Tutorial_14.pdf - ESO 208A Computational Methods in Engineering Tutorial 14 Partial Differential Equations 1 Consider the non-dimensional form of 1-D

Tutorial_14.pdf - ESO 208A Computational Methods in...

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ESO 208A: Computational Methods in Engineering Tutorial 14 Partial Differential Equations 1. Consider the non-dimensional form of 1-D diffusion equation 1 0 where 2 2 x x C t C , with constant boundary conditions; C (0, t ) = C (1, t ) = C 0 and the initial condition : 2 2 2 2 2 1 0 ,  x e x C ; with = 0.5 & = 0.5 (Gaussian narrow enough, such that the initial condition satisfies the boundary conditions at x = 0 and 1, for all practical purposes). (i) Write the discretized form of the diffusion equation using an explicit scheme with forward difference in time and 2 nd order central difference in spatial domain. (ii) Use five uniformly spaced points for discretization of the spatial domain. Compute the maximum value of t that can be used for stable solution. Use a t equal to half of the maximum value and calculate the concentration profile C ( x , t ) after one time step with C 0 =0. Graphically show the expected trends of solutions as time progresses.
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