As an additional difficulty, most of the important processes in the atmosphere or oceans interact in a multitude of scales. Representing the smallest influential spatial or time scales in a global domain poses severe demands on computational resources as well as mathematical tools for accurate representation.

In this presentation, we will introduce a few examples of such multi-scale conservation laws, derived from applications such as volcano ash transport or tsunami wave dispersion. We will list requirements to discretization schemes for accurate representation of the underlying conservation principles, and give alternative structure-preserving discretizations, solving the above mentioned problems. With such carefully designed mathematical methods implemented in modern algorithms scientific computing - as a sub-branch of mathematical sciences - can help prepare for and mitigate natural disaster.

While first- and second-order exponential integrators are well understood, higher-order methods are difficult to construct. We demonstrate a technique that can circumvent this problem, converting classical integration methods into corresponding exponential discretizations.

Stiff-order conditions for exponential integrators were developed in a Banach space framework by Hochbruck and Ostermann [2005]. They showed that in the worst case, the well-known ETDRK4 integrator of Cox and Matthews can exhibit an order reduction from four to two. Although they speculated that a four-stage exponential integrator with stiff-order four does not exist, we have nevertheless succeeded in constructing a four-stage exponential version of the classical RK4 integrator with stiff-order four. Our ERK4 integrator performs well in numerical tests and generalizes to the case where the dependent variable is a vector and the linear operator is a diagonal matrix. We plan to extend our construction to the case where the linear operator is a general square matrix. We also wish to derive embedded exponential Runge-Kutta pairs that efficiently generate both a high- and low-order estimate, allowing dynamic adjustment of the time step to achieve a specified accuracy.

We will review shell models and discuss some of the issues of structure-preserving discretizations of shell equations.

In this talk, we introduce as an alternative low regularity Fourier integrators. They are obtained from Duhamel's formula in the following way: first, a Lawson-type transformation eliminates the leading linear term and second, the dominant nonlinear terms are integrated exactly in Fourier space. For nonlinear Schrödinger equations, first order convergence of such methods only requires the boundedness of one additional derivative of the solution, and second-order convergence the boundedness of two derivatives. This allows us to impose lower regularity assumptions on the data. Numerical experiments underline the favorable error behavior of the newly introduced integrators for low regularity solutions compared to classical splitting and exponential integration schemes.

This is joint work with Marvin Knöller and Katharina Schratz (KIT, Karlsruhe).

The degrees of freedom are associated to geometric objects, and a relation to differential forms will be remarked. We show how to construct discrete polynomial function spaces of arbitrary degree associated to these geometric degrees of freedom, constituting a discrete de Rham complex: \[ \mathbb{R} \longrightarrow V^{0}_{h}\subseteq H(\nabla,\Omega) \stackrel{\nabla}{\longrightarrow} V^{1}_{h}\subseteq H(\nabla\times,\Omega) \stackrel{\nabla\times}{\longrightarrow} V^{2}_{h}\subseteq H(\nabla\cdot,\Omega) \stackrel{\nabla\cdot}{\longrightarrow} V^{3}_{h}\subseteq L^{2}(\Omega) \stackrel{}{\longrightarrow} 0\,. \] In this way it is possible to exactly discretize topological equations even on highly deformed meshes.

An important aspect in the numerical solution of partial differential equations is the efficient construction of the system matrix and its subsequent solution. Given a polynomial basis $\Psi_{i}$ which spans the polynomial vector space $\mathcal{P}$, this work addresses the construction and use of the algebraic dual space $\mathcal{P}'$ and its canonical basis. It will be shown that a primal-dual formulation using these algebraic dual basis results in a very sparse system matrix where two of the sub-matrices consist of only incidence matrices. The method will be applied to a Dirichlet and Neumann problem and it is shown that the finite dimensional approximations satisfy $\phi^{h} = \nabla\cdot\boldsymbol{q}^{h}$ on any grid. The dual method is also applied to a constrained minimization problem, which leads to a mixed finite element formulation. The discretization of the constraint and the Lagrange multiplier will be independent of the grid size, grid shape and the polynomial degree of the basis functions.