Matlab - How to implement backward Euler's method?
 

<p>I am trying to implement these formulas:</p>

<h3>Forward Euler's method:</h3>

<p><a href="https://i.stack.imgur.com/8myS0.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8myS0.png" alt="enter image description here"></a></p>

<p>this is what I have tried:</p>

<pre class="lang-matlab prettyprint-override"><code>x_new = (speye(nv)+ dt * lambda * L) * x_old;
</code></pre>

<p>Is there anything wrong with this? How can I calculate this using <strong>sparse</strong> operation?</p>

<h3>Backward Euler's method:</h3>

<p><a href="https://i.stack.imgur.com/8kIu3.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8kIu3.png" alt="enter image description here"></a></p>

<p>I have tried this:</p>

<pre class="lang-matlab prettyprint-override"><code>x_new = (speye(nv)- dt * lambda * L) \ x_old;
</code></pre>

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

<hr>

<p><code>L</code> is a sparse matrix like this:</p>

<pre class="lang-matlab prettyprint-override"><code>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
</code></pre>

<p>also for other variable we have something like this:</p>

<pre class="lang-matlab prettyprint-override"><code>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]
</code></pre>



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