The Method of Lines Solution of Partial Differential Equations
The Method of Lines Solution of Partial Differential Equations
James M Hyman
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0001791688853 using Gear's method in time and fourth order centered differences in space on a 64 point uniform mesh. -62- Figure 5. 2 is the graph of the PDE solution at various times. The steady state solution is reached at approximately t = 0, 05, It is clear from Figure 5. 2 that the solution varies rapidly in the region near x = 1. Therefore, the coordinate transformation m. Ethod is indicated. The transformation z = X, m >^ 1, stretches the boundary layer region in a neighborhood of x = 1 .... The transformed equation is 1 2 _2 (5. 12) u^ - (mz ) u + m(m-l)z u - n sinh nu t zz z with boundary conditions u(0, t) =, u(l, t) = 1 . Equation (5. 12) reduces to (5. 11) for m = 1 as it should. The equation v;as solved for n = 10 and m = 1, 2, ... , 30 and in each case the steady state was determined. The table below lists the maximum, deviation of these steady state solutions from the exact solution (5. 10). m 1 2 3 4 5 7 10 13 15 20 25 30 Maximum Error 0, . 055 0, . 035 0, . 025 0, . 019 0, .
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