## Applied Mathematics

A basic principle of science is to create understanding by modeling and simulation.

In exact sciences the models often take the form of (stochastic-, partial-) differential or integral equations. Today, a great deal of models can be “solved” on the computer and models are analyzed by computer simulation. Modern science is computational science and Computational Mathematical Modeling is the modern manifestation of the basic principle of science.

At the Department of Mathemaics and Computer Science, we continuously work to make new, challenging problems computable, for example: High resolution shock simulations. Shock waves are characterized by an abrupt, discontinuous change in the characteristics of a flow. Therefore, concepts from classical numerical analysis, which typically rely on the smoothness of the solution, are not applicable. Instead, the physics of the flow must be captured correctly. That is why the Riemann problem is an essential building block in shock capturing algorithms. It is a challenge to design robust methods that can deal with strong shocks and at the same time are sensitive enough to resolve fine structures in complex flow patterns.

Another area of research is the simulation of uncertain dynamic processes like f. ex. stock prices or tomorrows weather. Uncertain and random effects in a dynamic system are modeled by stochastic noise. By numerical simulation one can predict the general trends and their fluctuations. Again the challenge is to simulate non-smooth evolutions. Research at IMADA focuses on the construction of high-order accurate numerical methods for stochastic differential equations. Combining forces we also work on numerical methods for stochastic conservation laws modeling f. ex. reacting flows with uncertainties.

Computational Mathematics also contributes to the modeling process itself, for example: Self calibrating models in computational geosciences. The problem is that over geological time scales crucial material parameters are very difficult to quantify. How should one estimate transport coefficients in sediments that were deposited millions of years ago? This question is a headache for the geologists trying to assess the chances of finding oil and gas in new regions. Applying modern software and intelligent mathematics we recently designed a self-calibrating depositional algorithm based on accessible data from wells: Numerical simulations become an integrated tool in the modeling process!

Computational Mathematics plays a key role in understanding the complex physical models describing Nature, from the smallest possible scales of particle physics to cosmological models of the Universe. The dynamics of such elaborate models can only be "solved" using complex numerical simulations, usually requiring large High-Performance-Computing facilities. Understanding the strong interactions of elementary particles' field will allow to discover new forces of Nature, besides the four we already know: gravity, electromagnetism and the weak and strong force. Thanks to numerical simulations, models of new physics can be understood and put to the test.

Computational Mathematical Modeling at SDU in Odense maintains active collaboration with: