Masters Projects in Scientific Computing

By

Jan Nordström

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PROJECT NO 1: OCEAN CIRKULATION CAUSED BY

THE TIDE ?

• Need efficient solver to compute the flow over a rough surface.

• We develop a finite difference code for a model problem.

• Good to know: mathematics, fluid mechanics, coding.

Seamounts Trences

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Schematic tasks in Project 1

  Consider a model equation for the Navier-Stokes equations.

  Transform to curvilinear coordinates to capture geometry.   Show well-posedness, derive boundary conditions.

  Discretize using finite differences, to obtain an algorithm.   Show stability of the algorithm using the energy method.

  Write a program, in Matlab.

  Run the program and analyse the results.   Write a report.

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PROJECT NO 2: WAVE PROPAGATION RELATED TO

EARTH QUAKES

• Need efficient solver to compute wave propagation in the ground.

• We rasie the order of accuracy on our finite difference scheme.

• Good to know: mathematics, wave propagation, coding.

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Schematic tasks in Project 2

  Consider a wave propagation equation on first order form.

  Show well-posedness, derive boundary conditions.   Discretize using finite differences, to obtain an algorithm.

  To approximate derivatives, use new and old operators.   Show stability of the algorithm using the energy method.

  Write a program, in Matlab.

  Run the program and analyse/compare the results.   Write a report.

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PROJECT NO 3: STABILIZE FINITE VOLUME SCHEMES

IN AERONAUTICAL APPLICATIONS

• The aeronatical industry use unstructured finite volume methods.

• We will stabilize a simplified model problem using matrix algebra.

• Required knowledge: mathematics, linear algebra, coding.

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Schematic tasks in Project 3

  Consider a hyperbolic equation on first order form.

  Show well-posedness, derive boundary conditions.   Discretize using finite volume method, to obtain an algorithm.

  Analyze matricies, split in symmetric and skew-symmetric form.   Construct an artificial dissipation based on the analysis.

  Show stability of the algorithm using the energy method.

  Check the accuracy after inserting artificial dissipation.   Write a program, in Matlab.

  Run the program and analyse the results.   Write a report.

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PROJECT NO 4: STOCHASTIC ANALYSIS OF HYPERBOLIC PROBLEMS

• How do unceartainties in data propagate to the answer ?

• We compute answers in terms of expected value and variance.

• Good to know: mathematics, stochastic processes, coding.

Where is the shock ?

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Schematic tasks in Project 4

  Consider the advection-diffusion equation with unceartainties.

  Derive bounds on the solution using PDE theory.   Expand the solution in the stochastic variable.

  A Galerkin projection will give deterministic system for answer.   Discretize using finite differences, to obtain an algorithm.

  Show stability of the algorithm using the energy method.

  Write a program, in Matlab.   Run the program and analyse the result (expectation, variance).

  Compare with the result obtained using PDE theory.   Write a report.

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PROJECT NO 5: STUDY OF SEPARATED FLOWS

• Separation is a very delicate process to simulate numerically.

• Our weak boundary conditions need to be evaluated.

• Good to know: mathematics, fluid mechanics, coding.

Separated flow No separation

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Schematic tasks in Project 5

  Learn how weak boundary conditions work in a modelproblem.

  Vary penalty strength to obtain more accuracy (stiffness ?).   Learn how to run NS3D, a code that solves the N-S equations.

  Learn how to make curvilinear meshes.   Run NS3D, analyse and plot the answer.

  Vary penalty strength, check accuracy. Draw conclusions.

  Write a report.

www.liu.se

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