## mercoledì 15 giugno 2016

### Week 2-3: general exponential schemes

During these two weeks I implemented the two schemes for a general exponential Runge-Kutta and Rosenbrock integrator, see the schemes described in my second post.
I also implemented another file for exponential Runge-Kutta integrators based on the following scheme.
Given the problem
$$u'(t) = F(t,u) = A u(t) + g(t, u(t)), \\ u(t_0) = u_0,$$
the new scheme is
$$U_{ni} = u_n + c_i h_n \varphi_1(c_i h_n A) F(t_n,u_n) + h_n \sum_{j=2}^{i-1} a_{ij}(h_n A) D_{nj} , \\ u_{n+1} = u_n + h_n \varphi_1(h_n A) F(t_n,u_n) + h_n \sum_{i=2}^s b_i(h_n A) D_{ni} ,$$
where
$$D_{nj} = g(t_n + c_j h_n, U_{nj}) - g(t_n, u_n).$$
The main motivation for this reformulation is that the vectors $D_{nj}$ are expected to be small in norm so the application of matrix functions to these vectors are more efficient.

I applied the general pattern to the third-order exponential Runge-Kutta method with tableau

where $\varphi_{j,k} =\varphi_j (c_k h A)$ and $\varphi_j =\varphi_j (h A)$.

Then I also applied the general Rosenbrock pattern to the fourth-order exponential method with a third-order error estimator. Its coefficients are

where $\varphi_j =\varphi_j (h J_n)$.

I tested the correctness and order of the schemes with the following example, a semilinear parabolic problem
$$\frac{\partial U}{\partial t} (x,t) - \frac{\partial^2 U}{\partial x^2} (x,t) = \frac{1}{1 + U(x,t)^2} + \Phi (x,t)$$
for $x \in [0,1]$ and $t \in [0,1]$ with homogeneous Dirichlet boundary conditions. $\Phi$ is chosen in such a way that the exact solution of the problem is $U(x,t) = x(1-x)e^{t}$. I discretize this problem in space by standard finite differences with 200 grid points.

For details see [HO 10] and [HO 05].

Within this week I will insert on Bitbucket the codes of phi functions and the three general schemes .