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pdes for electrostatics & heat flow 451100V(x, y)0–10040302010x0010 20 30 40yFigure 17.6 Left: A Gnuplot visualization of the computed electric potential for a capacitorwith finite width plates. Right: An OpenDX visualization of the charge distribution along oneplate determined by evaluating ∇ 2 V(x, y) (courtesy of J. Wetzel). Note the “lightening rod”effect of charge accumulating at corners and points.3. Repeat the process for different step sizes ∆ and draw conclusions regardingthe stability and accuracy of the solution.4. Once your program produces reasonable solutions, modify it so that itstops iterating after convergence is reached, or if the number of iterationsbecomes too large. Rather than trying to discern small changes inhighly compressed surface plots, use a numerical measure of precision,for example,trace = ∑ i|V[i][i]|,which samples the solution along the diagonal. Remember, this is a simplealgorithm and so may require many iterations for high precision. You shouldbe able to obtain changes in the trace that are less than 1 part in 10 4 . (Thebreak command or a while loop is useful for this type of test.)5. Equation (17.31) expresses the successive overrelaxation technique in whichconvergence is accelerated by using a judicious choice of ω. Determine bytrial and error the approximate best value of ω. You will be able to double thespeed.6. Now that the code is accurate, modify it to simulate a more realistic capacitorin which the plate separation is approximately 110of the plate length.You should find the field more condensed and more uniform between theplates.7. If you are working with the wire-in-the-box problem, compare your numericalsolution to the analytic one (17.18). Do not be surprised if you need tosum thousands of terms before the analytic solution converges!−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 451