How to implement backward Euler's method?

I am trying to implement these formulas:

Forward Euler's method:

this is what I have tried:

x_new = (speye(nv)+ dt * lambda * L) * x_old;

Is there anything wrong with this? How can I calculate this using sparse operation?

Backward Euler's method:

I have tried this:

x_new = (speye(nv)- dt * lambda * L) \ x_old;

How to implement the backward part where the existing x is calculated based on the new x? Is it OK to use division?

=============================================================================

L is a sparse matrix like this:

full(L) =

   -1.0000    0.2500    0.2500    0.2500    0.2500
    0.3333   -1.0000    0.3333         0    0.3333
    0.3333    0.3333   -1.0000    0.3333         0
    0.3333         0    0.3333   -1.0000    0.3333
    0.3333    0.3333         0    0.3333   -1.0000

also for other variable we have something like this:

nv = 5;
dt = 0.01;
lambda = 0.5;

x_old =   [-4 0 5;
            1 -5 5;
            1 0 1;
            1 5 5;
            1 0 0]

Answer 1

[Dinesh Yadav]

Hi,

Kindly go through the following existing posts for backward euler method.

https://www.mathworks.com/matlabcentral/answers/260047-solve-ode-using-backward-euler-s-method

https://www.mathworks.com/matlabcentral/fileexchange/31427-1st-oder-ode-solver-euler-backward-method-based

Hope it helps.