Looking for a Master's project? See below.
Most of my research is, one way or another, connected to compressible flows and aerodynamics governed by the Euler and Navier-Stokes equations.
Numerical analysis (linear): A mathematical model that aspires to predict future events has to be well-posed. Hence, well-posedness of time-dependent Partial Differential Equations (PDEs) is a main theme. By using the energy method, boundary conditions are deduced and an a priori on the solution derived. The analysis is then mimicked for semi-discrete approximations satisfying a Summation-by-parts (SBP) property and with boundary conditions enforced by Simultataneous Approximation Terms. This leads to stable schemes.
Another key question is the convergence rate of SBP-SAT schemes. The order of accuracy is easily derived for finite difference schemes, but it drops near boundaries for high-order schemes. However, the convergence rate is often higher than the worst case scenario and the question is: What are the weakest conditions for an improved convergence rate?
(The work on linear PDEs is not limited to aerodynamics but generally applicable to a large class of linear time-dependent PDEs.)
Numerical analysis of non-linear conservation laws: The overarching goals are the same as for linear PDEs. However, the mathematical theory is much less developed and well-posedness is largely unknown. This complicates the development of numerical schemes. One way to obtain some a priori bounds is to enforce entropy stability on the scheme. (A kind generalization of the energy stability in the linear case.) My work on such schemes has been aimed towards: extending them to high-order of accuracy; including boundary conditions in the entropy stability framework; enforce both entropy stability as well as additional bounds. The long term goal is to have a scheme that satisfies strong enough bounds to allow convergence to weak solutions.
Connected to this work is also the study of various alternative regularizations of compressible flow models.
High performance computing: To demonstrate the efficacy of the schemes it is necessary to run simulations of realistic problems. Such problems invariably requires large-scale computing facilities. Hence, the schemes are coded (in F90 and MPI) and run on parallel computers.
Master's projects: I can offer Master's projects related to the above themes. The tools of the first theme (linear numerical analysis) can easily be applied to various different PDEs. (Not only flow models.) That is, if you have a particular interest in a certain area of science that involves a time-dependent PDE, a possible project could be to carry out the energy analysis for the equation (leading to well-posedness) and then apply the SBP-SAT framework to mimic the estimate for the numerical scheme. The scheme is then coded and run to demonstrate that it works. Once finished, you will know how to analyze a PDE and define a well-posed problem. Furthermore, you will know how to use state-of-the-art numerical methods to solve a PDE.
If you are interested, you are most welcome to come and have a chat with me.
- 2017. Response to “Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation”. Journal of Scientific Computing. Published ahead of print: 1-5. doi: 10.1007/s10915-017-0468-x
- 2015. Weak solutions and convergent numerical schemes of modified compressible Navier-Stokes equations. Journal of Computational Physics. 288: 19-51. doi: 10.1016/j.jcp.2015.02.013
- 2015. Efficiency benchmarking of an energy stable high-order finite difference discretization. AIAA Journal. 53: 1845-1860. doi: 10.2514/1.J053500
- 2014. A note on ∞ bounds and convergence rates of summation-by-parts schemes. BIT Numerical Mathematics. 54: 823-830. Published 2014-02-04. doi: 10.1007/s10543-014-0471-7
- 2014. Review of summation-by-parts schemes for initial-boundary-value problems. Journal of Computational Physics. 268: 17-38. doi: 10.1016/j.jcp.2014.02.031
- 2014. Entropy-stable schemes for the Euler equations with far-field and wall boundary conditions. Journal of Scientific Computing. 58: 61-89. doi: 10.1007/s10915-013-9727-7
- 2012. Higher-order finite difference schemes for the magnetic induction equations with resistivity. IMA Journal of Numerical Analysis. 32: 1173-1193. doi: 10.1093/imanum/drq030
- 2011. Implicit-explicit schemes for flow equations with stiff source terms. Journal of Computational and Applied Mathematics. 235: 1564-1577. doi: 10.1016/j.cam.2010.08.015
- 2010. On stability of numerical schemes via frozen coefficients and the magnetic induction equations. BIT Numerical Mathematics. 50: 85-108. doi: 10.1007/s10543-010-0249-5
- 2009. Higher order finite difference schemes for the magnetic induction equations. BIT Numerical Mathematics. 49: 375-395. doi: 10.1007/s10543-009-0219-y
- 2009. A stable and conservative high order multi-block method for the compressible Navier-Stokes equations. Journal of Computational Physics. 228: 9020-9035. doi: 10.1016/j.jcp.2009.09.005
- 2009. A hybrid method for unsteady inviscid fluid flow. Computers & Fluids. 38: 875-882. doi: 10.1016/j.compfluid.2008.09.010
- 2009. Shock Capturing Artificial Dissipation for High-Order Finite Difference Schemes. Journal of Scientific Computing. 39: 454-484. doi: 10.1007/s10915-009-9285-1
- 2008. Stable and accurate schemes for the compressible Navier-Stokes equations. Journal of Computational Physics. 227. doi: 10.1016/j.jcp.2007.10.018
- 2008. An accuracy evaluation of unstructured node-centred finite volume methods. Applied Numerical Mathematics. 58: 1142-1158. doi: 10.1016/j.apnum.2007.05.002
- 2008. A stable high-order finite difference scheme for the compressible Navier-Stokes equations, no-slip boundary conditions. Journal of Computational Physics. 227: 4805-4824. doi: 10.1016/j.jcp.2007.12.028
- 2007. High order accurate computations for unsteady aerodynamics. Computers & Fluids. 36: 636-649.
- 2007. A stable high-order finite difference scheme for the compressible Navier-Stokes equations, far-field boundary conditions. Journal of Computational Physics. 225: 1020-1038.
- 2016. Entropy Stable Summation-by-Parts Formulations for Compressible Computational Fluid Dynamics. Chapter 19, pages 495-524. In:
- 2016. Handbook of Numerical Analysis, Volume 17. Elsevier. 641 pages. ISBN: 978-0-444-63789-5.
- 2013. On entropy stable boundary conditions for the Navier-Stokes equations. Article, pages 297-307. In:
- 2013. MekIT'13 Seventh National Conference on Computational Mechanics, Trondheim 13-14 May 2013. Akademika forlag. 308 pages. ISBN: 978-82-321-0266-2.