Set up the forward and backward Euler methods for the follow

Set up the forward and backward Euler methods for the following problem, give the tridiagonal system for each step of the backward method.

Solution

Step1:

implicit methods are more expensive to be implemented for non-linear problems since y_n+1 is given only in terms of an implicit equation. The implicit analogue of the explicit FE method is the backward Euler (BE) method. This is based on the following Taylor series expansion

equation 1:

Y_n= y(t _n+1-h) =y(t_n+1) –h dy/dt /t _n+1 +O(h²)

Equation 2:

Y_n+1= Y_n + hf(Y_{n+1 ,} t_{n +1})---- (implict ) backward euler method

step2:

i have assume U in terms of Y function

note that in Eq. 2, f (y_n+1,t_n+1) is not known, hence it gives us an implicit equation for the computation of y_n+1 .

For instance, let y_t(t,x)= 4y_xx(t,x)+1 This means that to obtain y_n+1, we need to solve the non-linear equation  at any given time step n. A suitable root finding technique such as the Newton-Raphson method can be used for this purpose. This is evidently much more time consuming than the explicit FE method where, for the problem above,

Step 3:

we have . y_t(t,x)= 4y_xx(t,x)+1

Well, why do we resort to implicit methods despite their high computational cost? The reason is that implicit techniques are stable.

Let\'s examine this for the same linear test problem we considered in the context of the FE method: dy/dt = 4 y, y(0) = 1.

Step4: In the case of linear problems, using BE is as easy as using FE, applying Eq. 2, substitute above values and get y_t+1

y_t+1 = y_t / 1+4x