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A weak boundary procedure for high order finite difference approximations of hyperbolic problems
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
2011 (English)Report (Other academic)
Place, publisher, year, edition, pages
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2011-019
National Category
Computational Mathematics
URN: urn:nbn:se:uu:diva-159353OAI: oai:DiVA.org:uu-159353DiVA: diva2:444411
Available from: 2011-09-23 Created: 2011-09-28 Last updated: 2011-11-04Bibliographically approved
In thesis
1. Weak Boundary and Interface Procedures for Wave and Flow Problems
Open this publication in new window or tab >>Weak Boundary and Interface Procedures for Wave and Flow Problems
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, we have analyzed the accuracy and stability aspects of weak boundary and interface conditions (WBCs) for high order finite difference methods on Summations-By-Parts (SBP) form. The numerical technique has been applied to wave propagation and flow problems.

The advantage of WBCs over strong boundary conditions is that stability of the numerical scheme can be proven. The boundary procedures in the advection-diffusion equation for a boundary layer problem is analyzed. By performing Navier-Stokes calculations, it is shown that most of the conclusions from the model problem carries over to the fully nonlinear case.

The work was complemented to include the new idea of using WBCs on multiple grid points in a region, where the data is known, instead of at a single point. It was shown that we can achieve high accuracy, an increased rate of convergence to steady-state and non-reflecting boundary conditions by using this approach.

Using the SBP technique and WBCs, we have worked out how to construct conservative and energy stable hybrid schemes for shocks using two different approaches. In the first method, we combine a high order finite difference scheme with a second order MUSCL scheme. In the second method, a procedure to locally change the order of accuracy of the finite difference schemes is developed. The main purpose is to obtain a higher order accurate scheme in smooth regions and a low order non-oscillatory scheme in the vicinity of shocks.

Furthermore, we have analyzed the energy stability of the MUSCL scheme, by reformulating the scheme in the framework of SBP and artificial dissipation operators. It was found that many of the standard slope limiters in the MUSCL scheme do not lead to a negative semi-definite dissipation matrix, as required to get pointwise stability.

Finally, high order simulations of shock diffracting over a convex wall with two facets were performed. The numerical study is done for a range of Reynolds numbers. By monitoring the velocities at the solid wall, it was shown that the computations were resolved in the boundary layer. Schlieren images from the computational results were obtained which displayed new interesting flow features.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2011. 42 p.
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 862
weak boundary conditions, multiple penalty, finite difference methods, summation-by-parts, high order scheme, hybrid methods, MUSCL scheme, shocks, stability, energy estimate, steady-state, non-reflecting
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
urn:nbn:se:uu:diva-159440 (URN)978-91-554-8176-6 (ISBN)
Public defence
2011-11-07, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:15 (English)
Available from: 2011-10-14 Created: 2011-10-02 Last updated: 2011-11-09Bibliographically approved

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