[SOLVED] MA322-Lab 8

Description

1. Consider following BVP

with exact solution

y(x) = 1 + x − cosx − (1 + π/2)sinx.

Use second order scheme to complete following table

Finally plot exact solution and finite difference solution for h = 1/64.

2. Consider the following BVP

.

Consider the following finite difference scheme

.

and U₁ = U_N = 0, where

Compute the local truncation error of the above scheme and show that it is O(h⁴). Hence show that the scheme is fourth order accurate. Take f(x) = sin(x) so that the exact solution is u(x) = sin(x). Write a computer program to implement the above scheme. Solve the problem for N = 10,20,40,80,160,320 grid points and compute error in maximum norm and discrete L² norm in each case. Plot the error versus N on a log-log plot and verify the fourth order accuracy in both the norms.

3. Consider following BVP

with h = 1/3. If the exact solution is y(x) = 2e^x − x − 1, find the absolute errors at the nodal points using second order finite difference scheme.

• Solve the boundary value problem

with h = 0.25, by using central difference approximation to  and

1. central difference approximation to, ii. backward difference approximation to , iii. forward difference approximation to .

If the exact solution is y(x) = (e^10x–1)/(e¹⁰ − 1), compare the magnitudes of errors at the nodal points in the three methods.