Date of Award

2019

Document Type

Thesis

Degree Name

Master of Science in Engineering (MSE)

Department

Mechanical and Aerospace Engineering

Committee Chair

Sarma Rani

Committee Member

Kader Frendi

Committee Member

Sivaguru Ravindran

Subject(s)

Partial differential equations--Numerical solutions., Finite volume method.

Abstract

Numerical solution of partial differential equations using the finite volume method (FVM) entails calculating the fluxes of conservative variables at the cell faces.The fluxes are computed through the \reconstruction" of cell-centered variables to the cell face(s). "Reconstruction" is performed using variables and their gradients stored at the cell center located to the left and to the right of an interface, leading to two distinct states of the conservative variables at a face. The resulting discontinuity in interface flux presents a well-known conundrum in the FVM, particularly when solving hyperbolic governing equations. For the non-linear, hyperbolic Euler equations, Godunov recognized that the discontinuity in the left and right reconstructed states is qualitatively similar to the Riemann's initial-value problem that involves the advection of a scalar discontinuity. Godunov developed an \exact" Riemann solver that enables the calculation of the interface fluxes exactly. The exact solver, however,is computationally expensive and inefficient leading to the development of a large variety of "approximate" Riemann solvers. The best-known "approximate" Riemann solvers are the Roe's scheme, which is spatially 1st-order accurate, and the Roe-MUSCL scheme, which is a second-order extension of the former. Accordingly, the goal of this thesis is to develop a high-order flux calculation scheme that retains the basic structure of the Roe's interface flux. Roe's interface flux consists of two terms { a central-differenced (CD) flux, and a diffusive flux that ensures the solution remains monotonic. In this thesis, the high-order scheme is achieved by augmenting the order of accuracy of both these terms. For the CD flux term, two approaches are considered. The 1st involves replacing the second-order CD flux with a fourth- or sixth-order CD flux obtained from the cell-centered quantities located on either side of the interface. The second involves retaining the original form, but increasing the order of accuracy of the left and right reconstructed states that are inputs to the flux. The latter approach is also adopted for the diffusive flux. Third- and fifth-order Weighted Essentially Non-Oscillatory (WENO) schemes are used to reconstruct the left and right states at a cell face, so that the overall flux scheme may be referred to as a hybrid Roe-WENO scheme. The developed higher-order scheme is applied to a number of canonical 1-D and 2-D scalar advection cases. It is seen that a higher-order reconstruction of the states at a cell face performs better than the fourth- or sixth-order central differencing.

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