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Masters Projects in Scientific Computing
By
Jan Nordström
2
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
3
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.
4
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.
5
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.
6
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.
7
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 ?
9
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.
10
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
11
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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MARKERINGSYTA FÖR BILDER
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