Research at Computational Mathematics
Ambition
To develop mathematically based and provable convergent methods for solving timedependent partial differential equations governing physical processes
Main activities
High Order Finite Difference Methods (FDM)
We have developed summationbyparts 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 
Earthquake simulation related to Japananese tsunam 
Colaborator: 
Colaborator: 
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 timedependent 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: Viktor Linders
Finite Volume Methods (FVM)
We have reused the framework for the high order finite difference methods and derived summationbyparts operators for first and second derivatives as well as suitable artificial dissipation operators.
Unstructured mesh around high lift configuration 
Flow in the nose region of a Naca0012 wing profile 
Colaborator: 
Colaborator: 
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  Unstructured/structured mesh around Landers fault 
Colaborator:  Colaborator: 
Contact: Jan Nordström
Multi Physics Coupling
 We develop wellposed and stable numerical coupling procedures for multiple equations sets. Lately we have considered the NavierStokes and the heat equation, fluid structure interaction and the elastic wave equation with friction laws.
FSI, low accuracy misses eigenfrequency  Conjugate heat transfer between air and gold 
Colaborator:  Colaborator: 
Contact: Jan Nordström
Boundary Conditions
 We analyze existing techniques and derive new formulations. Lately we have focused on solid wall boundary conditions for the NavierStokes equations.
NavierStokes solution close to solid boundary  Weak enforcement of solid wall boundary conditions 
Colaborator:  Colaborator: 
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: Hannes Frenander
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 Haarwavelets 
Unceartainty in density for the Sod test case, exact solution 
Colaborator:
Uncertainty Quantification (UQ) Laboratory at Stanford University 
Colaborator:
Uncertainty Quantification (UQ) Laboratory at Stanford University 
Contact: Jan Nordström
Fluid Mechanics of Vortices
 By using the energy method on the constant coefficient compressible NavierStokes equations we can show why vortices decay slowly in viscous flows
Velocity field in a vortex  Energy deacy for different types of flowfields 
Colaborator: 
Contact: Jan Nordström
Propagation of nerv signals in living tissue
 By solving the cable equation in combination with the HodgkinHuxley's equations we can simulate the propagation of nerve signals in the denditric tree.
Nerves and somas  Sketch of a nerve connecting to a soma 
Colaborator:
Department of Aeronautics and Astronautics, Stanford University 
Colaborator:
Department of Aeronautics and Astronautics, Stanford University 
Figure 1: Illustration of lateral heat conduction. Measurements at x = L are used to calculate the temperature at the surface x = 0.

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 search 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 illposed 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.
Contact: Fredrik Berntsson.
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Last updated: Mon Nov 14 08:17:57 CET 2016