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matlabiterationsparse-matrix
Hadi GhahremanNezhad
Hadi GhahremanNezhad

Reputation: 2445

Matlab - How to implement backward Euler's method?

I am trying to implement these formulas:

Forward Euler's method:

enter image description here

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:

enter image description here

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]

Upvotes: 0

Views: 1123

Answers (1)

KUTlime
KUTlime

Reputation: 8007

I don't see the domain here but the Backward Euler method is a basic ordinary differential equation solver.

There are two approaches how to solve the situation when x_new stands on both sides of the equation.

1. Fixed point iteration

You use x_new_temp and you set x_new_temp as x_old for the first iteration and do few iterations with the Forward Euler formula. After the iterations perhaps limited by some difference between x_new_temp from iteration i and i-1, you set x_new as x_new_temp from the last iteration.

2. Solve non-linear equation with appropriate method, e.g. Newton-Raphson method

The Backward Euler formula dictates:

y_new = y_old + k*f(t,y_old)

We can transform this into:

y_old + k*f(t,y_old) - y_new = 0

This is a basic non-linear function which can be solved with any numerical method intended to solve this type of problem.

In your case when matrix are involved, I would go with the fixed point iteration.

Upvotes: 1

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