I am trying to test the accuracy of my code to a PDE with no numerical solution. I am using a Backwards time centred space finite difference method. I am calculating the error using;
$\frac{u_{4h} - u_{2h}}{u_{2h} - u_{h}} = 2^p + \mathcal{O}(h)$,
where $h$ is the step size, $p$ is the order of the method and $u$ is my approximation. I am calculating the error using different spatial/time steps and calculating p. My scheme should be $\mathcal{O}(\Delta t, \Delta x^2)$ and so I expect $p$ to be $1$ and $2$ respectively - this is what I get.
But my question is what is the best way to test the numerical accuracy of a solution without any know analytic solution? I am aware of the method of manufactured solution, but wanting to use something more similar to above. Ideally, I could plot the error against $\Delta t$ and $\Delta x^2$ on a log log plot where I would get a straight line.
I am quite new to numerical methods so there may be a very obvious answer, but I have tried searching for this before asking my question here. In particular, I am solving the three coupled PDEs;
$\frac{\partial{Q}}{\partial{t}} = -RaPra^2\theta - Pra^2Q + Pr\frac{\partial^2{Q}}{\partial{z}^2}, \ \ \ \ \ \ \ \ \ (1)$
$\frac{\partial{\theta}}{\partial{t}} = w - a^2\theta + \frac{\partial^2{\theta}}{\partial{z}^2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$
$ Q = -a^2w + \frac{\partial^2{w}}{\partial{z}^2}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$
where $Ra, Pr, a$ are constants. However, more general advice on convergence rates for problems with unknown solutions would be apperciated.