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Tulane University

1.
Cui, Shumo.
Well-balanced *Central*-*upwind* * Schemes*.

Degree: PhD, 2015, Tulane University

URL: https://digitallibrary.tulane.edu/islandora/object/tulane:27990

Flux gradient terms and source terms are two fundamental components of hyperbolic systems of balance law. Though having distinct mathematical natures, they form and maintain an exact balance in a special class of solutions, which are called steady-state solutions. In this dissertation, we are interested in the construction of well-balanced schemes, which are the numerical methods for hyperbolic systems of balance laws that are capable of exactly preserving steady-state solutions on the discrete level. We first introduce a well-balanced scheme for the Euler equations of gas dynamics with gravitation. The well-balanced property of the designed scheme hinges on a reconstruction process applied to equilibrium variables – the quantities that stay constant at steady states. In addition, the amount of numerical viscosity is reduced in the areas where the flow is in (near) steady-state regime, so that the numerical solutions under consideration can be evolved in a well-balanced manner. We then consider the shallow water equations with friction terms, which become very stiff when the water height is close to zero. The stiffness in the friction terms introduces additional difficulty for designing an efficient well-balanced scheme. If treated explicitly, the stiff friction terms impose a severe restriction on the time step. On the other hand, a straightforward (semi-) implicit treatment of the stiff friction terms can greatly enhance the efficiency, but will break the well-balanced property of the resulting scheme. To this end, we develop a new semi-implicit Runge-Kutta time integration method that is capable of maintaining the well-balanced property under the time step restriction determined exclusively by non-stiff components in the underlying equations. The well-balanced property of our schemes are tested and verified by extensive numerical simulations, and notably, the obtained numerical results clearly indicate that the well-balanced property plays an important role in achieving high resolutions when a coarse grid is used.

Subjects/Keywords: Central-upwind Schemes; Shallow Water Equations; Euler Equations; School of Science & Engineering; Mathematics; Ph.D

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Cui, S. (2015). Well-balanced Central-upwind Schemes. (Doctoral Dissertation). Tulane University. Retrieved from https://digitallibrary.tulane.edu/islandora/object/tulane:27990

Chicago Manual of Style (16^{th} Edition):

Cui, Shumo. “Well-balanced Central-upwind Schemes.” 2015. Doctoral Dissertation, Tulane University. Accessed June 26, 2019. https://digitallibrary.tulane.edu/islandora/object/tulane:27990.

MLA Handbook (7^{th} Edition):

Cui, Shumo. “Well-balanced Central-upwind Schemes.” 2015. Web. 26 Jun 2019.

Vancouver:

Cui S. Well-balanced Central-upwind Schemes. [Internet] [Doctoral dissertation]. Tulane University; 2015. [cited 2019 Jun 26]. Available from: https://digitallibrary.tulane.edu/islandora/object/tulane:27990.

Council of Science Editors:

Cui S. Well-balanced Central-upwind Schemes. [Doctoral Dissertation]. Tulane University; 2015. Available from: https://digitallibrary.tulane.edu/islandora/object/tulane:27990

Tulane University

2.
Cheng, Yuanzhen.
* Central*-

Degree: 2016, Tulane University

URL: https://digitallibrary.tulane.edu/islandora/object/tulane:72584

Shallow water models are widely used to describe and study fluid dynamics phenomena where the horizontal length scale is much greater than the vertical length scale, for example, in the atmosphere and oceans. Since analytical solutions of the shallow water models are typically out of reach, development of accurate and efficient numerical methods is crucial to understand many mechanisms of atmospheric and oceanic phenomena. In this dissertation, we are interested in developing simple, accurate, efficient and robust numerical methods for two shallow water models – the Saint-Venant system of shallow water equations and the two-mode shallow water equations. We first construct a new second-order moving-water equilibria preserving central-upwind scheme for the Saint-Venant system of shallow water equations. Special reconstruction procedure and source term discretization are the key components that guarantee the resulting scheme is capable of exactly preserving smooth moving-water steady-state solutions and a draining time-step technique ensures positivity of the water depth. Several numerical experiments are performed to verify the well-balanced and positivity preserving properties as well as the ability of the proposed scheme to accurately capture small perturbations of moving-water steady states. We also demonstrate the advantage and importance of utilizing the new method over its still-water equilibria preserving counterpart. We then develop and study numerical methods for the two-mode shallow water equations in a systematic way. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches – two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme – and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method for this system.

1

Yuanzhen Cheng

Subjects/Keywords: central-upwind schemes; 1-D shallow water equations; two-mode shallow water equations

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Cheng, Y. (2016). Central-Upwind Schemes for Shallow Water Models. (Thesis). Tulane University. Retrieved from https://digitallibrary.tulane.edu/islandora/object/tulane:72584

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Cheng, Yuanzhen. “Central-Upwind Schemes for Shallow Water Models.” 2016. Thesis, Tulane University. Accessed June 26, 2019. https://digitallibrary.tulane.edu/islandora/object/tulane:72584.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Cheng, Yuanzhen. “Central-Upwind Schemes for Shallow Water Models.” 2016. Web. 26 Jun 2019.

Vancouver:

Cheng Y. Central-Upwind Schemes for Shallow Water Models. [Internet] [Thesis]. Tulane University; 2016. [cited 2019 Jun 26]. Available from: https://digitallibrary.tulane.edu/islandora/object/tulane:72584.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Cheng Y. Central-Upwind Schemes for Shallow Water Models. [Thesis]. Tulane University; 2016. Available from: https://digitallibrary.tulane.edu/islandora/object/tulane:72584

Not specified: Masters Thesis or Doctoral Dissertation

Indian Institute of Science

3. Jaisankar, S. Accurate Computational Algorithms For Hyperbolic Conservation Laws.

Degree: 2008, Indian Institute of Science

URL: http://hdl.handle.net/2005/905

The numerics of hyperbolic conservation laws, e.g., the Euler equations of gas dynamics, shallow water equations and MHD equations, is non-trivial due to the convective terms being highly non-linear and equations being coupled. Many numerical methods have been developed to solve these equations, out of which central schemes and upwind schemes (such as Flux Vector Splitting methods, Riemann solvers, Kinetic Theory based Schemes, Relaxation Schemes etc.) are well known. The majority of the above mentioned schemes give rise to very dissipative solutions. In this thesis, we propose novel low dissipative numerical algorithms for some hyperbolic conservation laws representing fluid flows. Four different and independent numerical methods which give low diffusive solutions are developed and demonstrated.
The first idea is to regulate the numerical diffusion in the existing dissipative schemes so that the smearing of solution is reduced. A diffusion regulator model is developed and used along with the existing methods, resulting in crisper shock solutions at almost no added computational cost. The diffusion regulator is a function of jump in Mach number across the interface of the finite volume and the average Mach number across the surface. The introduction of the diffusion regulator makes the diffusive parent schemes to be very accurate and the steady contact discontinuities are captured exactly. The model is demonstrated in improving the diffusive Local Lax-Friedrichs (LLF) (or Rusanov) method and a Kinetic Scheme. Even when employed together with accurate methods of Roe and Osher, improvement in solutions is demonstrated for multidimensional problems.
The second method, a Central Upwind-Biased Scheme (CUBS), attempts to reorganize a central scheme such that information from irrelevant directions is largely reduced and the upwind biased information is retained. The diffusion co-efficient follows a new format unlike the use of maximum characteristic speed in the Local Lax-Friedrichs method and the scheme results in improved solutions of the flow features. The grid-aligned steady contacts are captured exactly with the reorganized format of diffusion co-efficient. The stability and positivity of the scheme are discussed and the procedure is demonstrated for its ability to capture all the features of solution for different flow problems.
Another method proposed in this thesis, a Central Rankine-Hugoniot Solver, attempts to integrate more physics into the discretization procedure by enforcing a simplified Rankine-Hugoniot condition which describes the jumps and hence resolves steady discontinuities very accurately. Three different variants of the scheme, termed as the Method of Optimal Viscosity for Enhanced Resolution of Shocks (MOVERS), based on a single wave (MOVERS-1), multiple waves (MOVERS-n) and limiter based diffusion (MOVERS-L) are presented. The scheme is demonstrated for scalar Burgers equation and systems of conservation laws like Euler equations, ideal Magneto-hydrodynamics equations and shallow water equations.…
*Advisors/Committee Members: Rao, S V Raghurama.*

Subjects/Keywords: Gas Dynamics; Magnetohydrodynamics; Conservation Laws; Algorithms; Numerical Analysis; Diffusion (Mathematical Physics); Hyperbolic Equations (Mathematical Analysis); Diffusion Regulator Model; Hyperbolic Partial Differential Equations; Compressible Flows - Numerical Methods; Hyperbolic Consevation Laws; Diffusion Regulated Schemes; Upwind-Biased Scheme; Rankine Hugoniot Solver; Grid-free Central Solver; Applied Mechanics

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Jaisankar, S. (2008). Accurate Computational Algorithms For Hyperbolic Conservation Laws. (Thesis). Indian Institute of Science. Retrieved from http://hdl.handle.net/2005/905

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Jaisankar, S. “Accurate Computational Algorithms For Hyperbolic Conservation Laws.” 2008. Thesis, Indian Institute of Science. Accessed June 26, 2019. http://hdl.handle.net/2005/905.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Jaisankar, S. “Accurate Computational Algorithms For Hyperbolic Conservation Laws.” 2008. Web. 26 Jun 2019.

Vancouver:

Jaisankar S. Accurate Computational Algorithms For Hyperbolic Conservation Laws. [Internet] [Thesis]. Indian Institute of Science; 2008. [cited 2019 Jun 26]. Available from: http://hdl.handle.net/2005/905.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Jaisankar S. Accurate Computational Algorithms For Hyperbolic Conservation Laws. [Thesis]. Indian Institute of Science; 2008. Available from: http://hdl.handle.net/2005/905

Not specified: Masters Thesis or Doctoral Dissertation