Statement of the problem
I try to investigate the order of accuracy in the case of using BTCS time stepping method applied to solving unsteady advection-diffusion equation using the finite difference method.
I try to investigate order of accuracy with respect to the mesh spacing and the time step. In other words, I assume that the error could be decomposed in the following way:
$$ error = O(\Delta x^{p}) + O(\Delta q^{q}) $$
where $p$, $q$ are some natural numbers to be determined
What is known
I know that BTCS scheme can be written as:
$$ \frac{\boldsymbol{c}^{n+1} - \boldsymbol{c}^{n}}{\Delta t} = A\boldsymbol{c}^{n+1} $$
where:
- $c^{n}$ refers to $n$-th state of the system in the computation
- matrix $A$ is defined as
$$ A := -\frac{U}{\Delta x}D_{x} + \frac{\kappa}{(\Delta x)^{2}}D_{xx} $$
where:
- $U$ refers to advection speed
- $\kappa$ refers to diffusivity constant
- $D_{x}$ refers to advective matrix
- $D_{xx}$ refers to diffusivity matrix
- $\Delta x$ refers to mesh spacing size
What I tried
I thought of applying the analysis similar to propagation error that is primarily done in processing experimental results in physics/engineering. However, I am doubtful about this approach.
I manipulated BTCS scheme and got alternative form: $$ c^{n + 1} = (I - \Delta t A)^{-1} c^{n} $$
I know that applying this equation $n$ times one may obtain the following equation:
$$ c^{n} = (I - \Delta t A)^{-n} c^{0} $$
However, I am not sure what to do next, i.e. I am not sure whether to apply Taylor series for multivariable function and proceed further.
Actually speaking, I do not have intuition how to proceed further.
From computer simulation I obtain linear dependency for $\Delta t$ and quadratic dependency for $\Delta x$. However, I may be wrong.
Any help how to proceed would be very appreciated!!!