Masters Projects in Scientific Computing. By Jan Nordström

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1 Masters Projects in Scientific Computing By Jan Nordström

2 PROJECT NO 1: OCEAN CIRKULATION CAUSED BY THE TIDE? Seamounts Trences 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. 2

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. 3

We rasie the order of accuracy on our finite difference

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. 4

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. 5

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. 6

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. 7

8 PROJECT NO 4: STOCHASTIC ANALYSIS OF HYPERBOLIC PROBLEMS Where is the shock? 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. 8

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. 9

10 PROJECT NO 5: STUDY OF SEPARATED FLOWS Separated flow No separation Separation is a very delicate process to simulate numerically. Our weak boundary conditions need to be evaluated. Good to know: mathematics, fluid mechanics, coding. 10

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. 11

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14 MARKERINGSYTA FÖR BILDER När du gör egna slides, placera bilder och andra illustrationer inom dessa fält. Titta gärna i baspresentationen för exempel på hur placeringen kan göras. 14

1.2 Numerical Solutions of Flow Problems

1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian