Numerical Solutions of Time-Dependent Partial Differential Equations

Unstructured mesh around high lift configuration

We have developed high order accurate and stable finite difference methods and improved the boundary treatment of finite volume methods. Another line of work deals with the development and implementation of boundary conditions for the Euler and Navier-Stokes equations. Future applications will include more coupled multi-physics problem. We are interested in well posedness for multiple equations sets as well as stable numerical coupling procedures. We consider various kinds of uncertainties in the data or parameters of the problem and aim for a computational methodology that delivers an answer with error bars.

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

High speed separated flow including shocklets

Collaborator: Flow Research Unit, University of the Witwatersrand

Earthquake simulation related to Japananese tsunam

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

A time varying deforming domain

The rate of dilatation

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

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

Pressure profile of propagating Euler vortex

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

Unstructured mesh around high lift configuration

Collaborator: FOI, Swedish Defence Research Agency

Flow in the nose region of a Naca0012 wing profile

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

Unstructured/structured mesh around rods

Collaborator: CTR, Center for Turbulence Research at Stanford University

Unstructured/structured mesh around Landers fault

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

FSI, low accuracy misses eigenfrequency

Collaborator: Department of Information Technology, Uppsala University

Conjugate heat transfer between air and gold

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

Navier-Stokes solution close to solid boundary

Collaborator: Department of Information Technology, Uppsala University

Weak enforcement of solid wall boundary conditions

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

En oscillerande puls som vi vill modellera

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

Felet som funktion av tiden med och utan MPT.

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

Unceartainty in density for the Sod test case

Collaborator: Uncertainty Quantification (UQ) Laboratory at Stanford University

Unceartainty in density for the Sod test case, exact solution

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

Velocity field in a vortex

Collaborator: Department of Information Technology, Uppsala University

Energy deacy for different types of flowfields

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

Nerves and somas

Collaborator: Department of Aeronautics and Astronautics, Stanford University

Sketch of a nerve connecting to a soma

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.

Illustration av sidledsvärmeledning

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

Defence of doctoral theses
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25 October 2019

Defence in Computational Mathematics: Andrea Alessandro Ruggiu

1.15 pm – 3.30 pm Ada Lovelace, B Building, entrance 27, ground floor, Campus Valla

Andrea Alessandro Ruggiu, at the Department of Mathematics, defends his doctoral thesis entitled "Eigenvalue analysis and convergence acceleration techniques for summation-by-parts approximations". Opponent is Daniel Appelö, Associate Professor, University of Colorado Boulder. The defence will be in English and is open to the public.

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Andrea Alessandro Ruggiu

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