Simplify multistep higher-order correction

In the multistep code, we start with a linear multistep method

$(M + \eta K)u_n = \text{RHS}(u_{n-1},\ldots,u_{n-k})$

We want to replace $M + \eta K$ with $\tilde{M} = (M_x + \eta K_x) \otimes (M_y + \eta K_y)$ to use ADS solver, but this introduces $O(\tau^2)$ error in each time step, which reduces the order of method to 1, making it pointless to go for higher order schemes.

To prevent that, we add some stuff to both sides so that $M + \eta K$ is applied to a small quantity. For example, $u_n - 2u_{n-1} + u_{n-2}$ is of order $O(\tau^2)$ , so we can construct a third order method starting with

$(M + \eta K)(u_n - 2u_{n-1} + u_{n-2}) = \text{RHS}(u_{n-1},\ldots,u_{n-k}) + (M + \eta K)(-2u_{n-1} + u_{n-2})$

and replacing $M + \eta K$ with $\tilde{M}$ :

$\tilde{M}(u_n - 2u_{n-1} + u_{n-2}) = \text{RHS}(u_{n-1},\ldots,u_{n-k}) + (M + \eta K)(-2u_{n-1} + u_{n-2})$

In the previous version of the code, we would move everything except for $u_n$ to the right-hand side and end up with

$\tilde{M}u_n = \text{RHS}(u_{n-1},\ldots,u_{n-k}) - \eta^2 K_x \otimes K_y (-2u_{n-1} + u_{n-2})$

since $M + \eta K - \tilde{M} = - \eta^2 K_x \otimes K_y$ . This is, however, a bad idea, since the bilinear form associated with $K_x \otimes K_y$ is unnatural and difficult to evaluate (requires mixed derivatives), especially in 3D. Instead, it is better to solve for $\mathcal{U} = u_n - 2u_{n-1} + u_{n-2}$ and compute $u_n$ from it.

Closes #59

marcinlos / iga-ads

Simplify multistep higher-order correction #62

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