Energy and entropy stable numerical methods with injected boundary conditions

I denne avhandlingen studerer vi de kompressible Navier-Stokes-likningene formulert med både adiabatiske veggrandvilkår og fjernfeltvilkår. Selv om det er ukjent om disse likningene er velformulerte er de av stor interesse, og de er mye brukt innen numerisk fluiddynamikk. Et resultat av Strang (1964) sier at for ikke-lineære problem diskretisert ved hjelp av en differansemetode som er lineærstabil, er denne metoden konvergent for glatte løsninger. Altså finnes det teori vi kan bruke i analysen av Navier-Stokes-likningene. Derfor studerer vi her teori for velformulerte lineære problem, og stabilitet for numeriske metoder. Dette gjøres både for de kompressible Navier-Stokes-likningene, men også for lineære partielle differensiallikninger som modellproblem. Videre utleder vi entropiestimat for de ikke-lineære Navier-Stokes-likningene, et estimat som virker som et kriterium for den svake løsningen vi leter etter; den skal i tillegg til likningene tilfredsstille termodynamikkens andre lov. Hovedfokuset ved dette arbeidet er stabil håndtering av de adiabatiske veggrandvilkårene og fjernfeltvilkår for Navier-Stokes-likningene. Vi beviser at heftelsesvilkåret (eng.: no- slip condition) kan bli implementert eksakt og fremdeles resultere i et entropiestimat når teknikken brukes i kombinasjon med delvissummasjonsoperatorer (SBP-operatorer) som har diagonale normmatriser og randmatriser. Vi introduserer også en ny metodikk for å sette fjernfeltvilkår, og beviser at den fører til et entropistabilt skjema for de kompressible Navier-Stokes-likningene. Teknikken er i tillegg lineært velformulert. Gjennom hele arbeidet bruker vi SBP-operatorer på grunn av deres gode stabilitetsegenskaper. Vi beviser også at en litt endret versjon av SBP-operatoren som tilnærmer den andrederiverte ved hjelp av endelig-volummetoden gitt av Chandrashekar (2016) er (svakt) konsistent, noe som gjør den egnet til å diskretisere de viskøse leddene i Navier-Stokes-likningene på ustrukturerte gitter.The compressible Navier-Stokes equation subject to both adiabatic wall boundary conditions and far-field boundary conditions are studied in this thesis. Although the well- posedness of these equations is generally unknown, they are of wide interest and are extensively used in computational fluid dynamics. A result by Strang (1964) states that if a non-linear problem is discretised using a difference method that is linearly stable, then this method is convergent for smooth solutions. That is, there exists theory we can use in the analysis of the Navier-Stokes equations. Thus, we study linear well-posedness and stability of numerical schemes both in the context of the compressible Navier-Stokes equations, but also linear partial differential equations as model problems. Furthermore, entropy estimates are derived for the fully non-linear Navier-Stokes equations, which pose as an admissibility criterion for the relevant weak solution we seek; it should additionally satisfy the second law of thermodynamics. The main focus of this work is the stable imposition of the adiabatic wall and far-field boundary conditions for the Navier-Stokes equations. In particular, we prove that the no-slip condition can be imposed strongly and still yield an entropy estimate when used in combination with diagonal-norm summation-by-parts (SBP) operators with diagonal boundary operators. Furthermore, we introduce a new methodology for setting far- field boundary conditions, and prove that it leads to an entropy stable scheme for the compressible Navier-Stokes equations. The procedure is additionally linearly well-posed. Throughout, we employ SBP operators due to their remarkable stability properties. We also prove that a slightly modified version of the finite-volume SBP approximation of the second-derivative given by Chandrashekar (2016) is (weakly) consistent, thus making it suitable for discretising the viscous terms of the Navier-Stokes equations on unstructured grids.Doktorgradsavhandlin

SBP-SAT schemes for hyperbolic problems

Numerical methods for solving partial differential equations is an important field of study, as it helps us to describe many different processes in the world. An important property of a numerical method, is that it should be a stable approximation of the governing differential equation. For numerical approximations that satisfy a summation-by-parts rule, and that are combined with the simultaneous approximation term technique at the boundaries, energy estimates can be derived to prove stability. The Summation-By-Parts Simultaneous Approximation Term (SBP-SAT) technique was first developed in the context of the finite difference method. More recently, it has been shown that other numerical methods, such as the finite volume method, also can be formulated in the SBP framework. The finite volume method is a popular numerical method, as it can be formulated on unstructured grids. However, Svärd et al. ([1]) showed that some approximations of the second derivative are in fact inconsistent on such grids. Consistency is another key feature of a numerical method. The method should be consistent in order for us to know that we are solving the correct equation. In this thesis, we study the extension of the SBP-SAT technique to the finite volume method. We introduce a methodology for implementing a second derivative approximation on general unstructured grids by including a transformation to a computational domain, where accuracy is expected to be recovered. The numerical experiments demonstrate that full accuracy is not obtained when including the transformation. There are still nodes along and near the boundary that are inconsistent. However, numerical experiments indicate that we have convergence. [1]: M. Svärd, J. Gong, and J. Nordström. An accuracy evaluation of unstructured node-centred finite volume methods. Applied Numerical Mathematics, 58:1142–1158, 2007. doi: 10.1016/j.apnum.2007. 05.002.Masteroppgave i anvendt og beregningsorientert matematikkMAB399MAMN-MA

Convergence of Chandrashekar’s Second-Derivative Finite-Volume Approximation

We consider a slightly modified local finite-volume approximation of the Laplacian operator originally proposed by Chandrashekar (Int J Adv Eng Sci Appl Math 8(3):174–193, 2016, https://doi.org/10.1007/s12572-015-0160-z). The goal is to prove consistency and convergence of the approximation on unstructured grids. Consequently, we propose a semi-discrete scheme for the heat equation augmented with Dirichlet, Neumann and Robin boundary conditions. By deriving a priori estimates for the numerical solution, we prove that it converges weakly, and subsequently strongly, to a weak solution of the original problem. A numerical simulation demonstrates that the scheme converges with a second-order rate.publishedVersio

Entropy stability for the compressible Navier-Stokes equations with strong imposition of the no-slip boundary condition

We consider the compressible Navier-Stokes equations subject to no-slip adiabatic wall boundary conditions. The main goal is to investigate stability properties of schemes imposing the no-slip condition strongly (injection) and the temperature condition weakly by a simultaneous approximation term. To this end, we propose a low-order summation-by-parts scheme. By verifying the complete linearisation procedure, we prove linear stability for the scheme. In addition, and assuming that the interior scheme is entropy stable, we also prove entropy stability for the full scheme including the boundary treatment. Furthermore, we propose a linearly stable 3rd-order scheme with the same imposition of the wall conditions. However, the 3rd-order scheme is not provably non-linearly stable. A number of simulations show that the boundary procedure is robust for both schemes.publishedVersio

SBP-SAT schemes for hyperbolic problems

Numerical methods for solving partial differential equations is an important field of study, as it helps us to describe many different processes in the world. An important property of a numerical method, is that it should be a stable approximation of the governing differential equation. For numerical approximations that satisfy a summation-by-parts rule, and that are combined with the simultaneous approximation term technique at the boundaries, energy estimates can be derived to prove stability. The Summation-By-Parts Simultaneous Approximation Term (SBP-SAT) technique was first developed in the context of the finite difference method. More recently, it has been shown that other numerical methods, such as the finite volume method, also can be formulated in the SBP framework. The finite volume method is a popular numerical method, as it can be formulated on unstructured grids. However, Svärd et al. ([1]) showed that some approximations of the second derivative are in fact inconsistent on such grids. Consistency is another key feature of a numerical method. The method should be consistent in order for us to know that we are solving the correct equation. In this thesis, we study the extension of the SBP-SAT technique to the finite volume method. We introduce a methodology for implementing a second derivative approximation on general unstructured grids by including a transformation to a computational domain, where accuracy is expected to be recovered. The numerical experiments demonstrate that full accuracy is not obtained when including the transformation. There are still nodes along and near the boundary that are inconsistent. However, numerical experiments indicate that we have convergence. [1]: M. Svärd, J. Gong, and J. Nordström. An accuracy evaluation of unstructured node-centred finite volume methods. Applied Numerical Mathematics, 58:1142–1158, 2007. doi: 10.1016/j.apnum.2007. 05.002