## Collaboration and funding

The research is done in collaboration with NASA Langley Research Center, Stanford University and Southern Methodist University in USA, the Council for Scientific and Industrial Research (CSIR) in South Africa, the Swedish Defence Research Agency (FOI) and the Swedish Meteorological and Hydrological Institute (SMHI) in Sweden.

The base funding is from Linköping University while the external research is funded by Swedish Research Council, SMHI and the European Union FP7 projects Industrialisation of High-Order Methods - A Top-Down Approach (IDIHOM) and Uncertainty Management for Robust Industrial Design in Aeronautics (UMRIDA).

## Our ambition

To develop mathematically based and provable convergent methods for solving time-dependent partial differential equations governing physical processes.

## Main activities:

## High Order Finite Difference Methods (FDM)

We have developed summation-by-parts operators and penalty techniques for boundary and interface conditions. We have applied FDM to demanding problems in fluid mechanics and wave propagation.

*High speed separated flow including shocklets*

Collaborator: Flow Research Unit, University of the Witwatersrand

*Earthquake simulation related to Japananese tsunam*

Collaborator: Department of Geophysics, Stanford University

**Contact: Jan Nordström**

We have extended the FDM to handle partial differential equations posed on time varying deforming domains. As an application, sound propagation by the linearized Euler equations in a time-dependent domain is considered.

*A time varying deforming domain*

*The rate of dilatation*

**Contact: Jan Nordström**

We have developed a method for constructing finite differences adjusted to periodic wave propagation problems. The resulting stencils minimise the wave speed error that arises in the discrete solution.

*Comparison of pulse propagation between the new Remez stencils and other stencils in the literature*

*Pressure profile of propagating Euler vortex*

**Contact: Jan Nordström**

## Finite Volume Methods (FVM)

We have reused the framework for the high order finite difference methods and derived summation-by-parts operators for first and second derivatives as well as suitable artificial dissipation operators.

*Unstructured mesh around high lift configuration*

Collaborator: FOI, Swedish Defence Research Agency

*Flow in the nose region of a Naca0012 wing profile*

Collaborator: FOI, Swedish Defence Research Agency

**Contact: Jan Nordström**

## Numerical Coupling

We combine the advantages of FDM and FVM in an efficient and stable way. This numerical technique has been used in both aerodynamics (the Euler equations) and wave propagation (elastic wave equations) problems.

*Unstructured/structured mesh around rods*

Collaborator: CTR, Center for Turbulence Research at Stanford University

*Unstructured/structured mesh around Landers fault*

Collaborator: Department of Geophysics, Stanford University

**Contact: Jan Nordström**

## Multi Physics Coupling

We develop well-posed and stable numerical coupling procedures for multiple equations sets. Lately we have considered the Navier-Stokes and the heat equation, fluid structure interaction and the elastic wave equation with friction laws.

*FSI, low accuracy misses eigenfrequency*

Collaborator: Department of Information Technology, Uppsala University

*Conjugate heat transfer between air and gold*

Collaborator: Department of Information Technology, Uppsala University

**Contact: Jan Nordström**

## Boundary Conditions

We analyze existing techniques and derive new formulations. Lately we have focused on solid wall boundary conditions for the Navier-Stokes equations.

*Navier-Stokes solution close to solid boundary*

Collaborator: Department of Information Technology, Uppsala University

*Weak enforcement of solid wall boundary conditions*

Collaborator: FOI, Swedish Defence Research Agency

**Contact: Jan Nordström**

SAT terms are used at the boundaries to implement the boundary conditions. However, if data is available, additional SAT terms can be applied in the interior of the domain. We refer to this technique as multiple penalty technique (MPT). By using this technique, we can reduce the errors in the computational domain and increase the rate of convergence.

*An oscillating pulse at different time levels that we want to model. Upper left: t=0, upper right: t=0.25, lower left: t=0.75 lower right: t=1.15*

*The error as a function of time when using 80 grid points in each space direction*

**Contact: Jan Nordström**

## Unceartainty Quantification (UQ)

We take into account various kinds of uncertainties or stochastic variations related to aerodynamic problems. Typical examples include stochastic uncertainties in the geometry of a wing, the speed of the aircraft and the angle of attack.

*Unceartainty in density for the Sod test case approximated using Haar-wavelets*

Collaborator: Uncertainty Quantification (UQ) Laboratory at Stanford University

*Unceartainty in density for the Sod test case, exact solution*

Collaborator: Uncertainty Quantification (UQ) Laboratory at Stanford University

**Contact: Jan Nordström**

## Fluid Mechanics of Vortices

By using the energy method on the constant coefficient compressible Navier-Stokes equations we can show why vortices decay slowly in viscous flows.

*Velocity field in a vortex*

Collaborator: Department of Information Technology, Uppsala University

*Energy deacy for different types of flowfields*

**Contact: Jan Nordström**

## Propagation of nerv signals in living tissue

By solving the cable equation in combination with the Hodgkin-Huxley's equations we can simulate the propagation of nerve signals in the denditric tree.

*Nerves and somas*

Collaborator: Department of Aeronautics and Astronautics, Stanford University

*Sketch of a nerve connecting to a soma*

Collaborator: Department of Aeronautics and Astronautics, Stanford University

**Contact: Jan Nordström**

## Heat Conduction

In industrial applications one would sometimes estimate the temperature in inaccessible areas where the temperature can not be measured directly. You then place a measuring element inside the material and calculate the sought surface temperature from the measured data. Mathematically, it is described as a Cachy problem for the heat equation with data available along the line x = L and the solution at x = 0 is requested. The situation is illustrated in Figure 1. The problem is ill-posed when small measurement errors can cause large errors in the computed solution. Special regularization methods are used to minimize the effect of measurement errors of calculated solutions. Similar mathematical problems arise in applications where mathematical analysis is used to compensate for errors caused by, for example, the measurement devices.

Research is done within the methodological and mathematical analysis of both the application problems and numerical methods.

Figure 1: Illustration of lateral heat conduction. Measurements at x = L are used to calculate the temperature at the surface x = 0.

**Contact: Fredrik Berntsson**