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Table of contents
1 Introduction
1.1 Background
1.2 Computational uid dynamics
1.3 Chorin’s projection method
1.4 Error of a numerical scheme
1.5 Basilisk
1.6 Grids
1.6.1 Basilisk adaptivity
1.7 Fields
1.8 Boundary Conditions
1.9 State of the art for Basilisk at the start of my PhD
1.10 Outline of this PhD Thesis
2 Advection Solver
2.1 Context
2.2 Godunov’s scheme and the Riemann problem
2.3 Temporal Scheme – Runge{Kutta schemes
2.4 Basilisk – State of the art: October 2015
2.5 Advection scheme by Bell, Colella and Glaz
2.5.1 Convection term
2.6 Weighted Essentially Non Oscillatory (WENO) scheme
2.6.1 Left and right side reconstruction
2.6.2 Fifth-order WENO { Formulations
2.6.3 Implementing WENO-5 stencils on Basilisk
2.6.4 Literature review
2.7 Test case { Advection in a 1D Domain
2.7.1 Passive advection of a 1D smooth tracer eld
2.7.2 Passive advection of a 1D discontinuous tracer eld
2.8 WENO in 2D and 3D cases
2.8.1 Transverse sweeps – Gaussian quadratures
2.9 Test case – Passive advection in a 2D domain
2.9.1 Periodic tracer in a uniform velocity eld
2.9.2 Compact tracer in a solid body rotation
2.10 Multi-resolution analysis
2.10.1 Wavelet Transform – Lifting Algorithm
2.10.2 Wavelet transform and Basilisk adaptivity
2.10.3 Restriction operator
2.10.4 Prolongation operator
2.10.5 Fifth-order prolongation
2.10.6 Testing the order of the prolongation operator
2.11 Advection of a tracer under rotation and stretching
2.11.1 Uniform grid computations
2.11.2 Adaptive grid computation for the tracer advection problem
2.12 Conclusion
3 Poisson{Helmholtz Solver
3.1 Context
3.2 State of the Art: October 2015
3.3 Numerical Algorithm { Poisson Solver
3.3.1 Iterative Methods
3.3.2 Multigrid Methods
3.3.3 Discretization Scheme { Second-order solver
3.3.4 Discretization Scheme { Fourth-order solver
3.3.5 Higher dimension cases
3.3.6 Boundary Conditions
3.4 Results for the 9-point stencil
3.4.1 Uniform grid { Direct problem
3.4.2 Uniform grid { Inverse problem
3.4.3 Non-uniform grid { Direct problem
3.5 Convergence studies on adaptive grids
3.6 Conclusion
3.6.1 Applications of the Poisson{Helmholtz solver
4 Navier{Stokes Solver
4.1 Governing equations
4.2 Literature survey
4.3 Navier{Stokes solver by Bell, Colela and Glaz
4.3.1 Temporal discretization
4.3.2 Projection Algorithm
4.3.3 Viscous dissipation terms
4.4 Higher-order method for Navier{Stokes equations
4.4.1 Time-marching schemes
4.4.2 Convection term – WENO interpolation and Riemann Solver
4.4.3 Projection Algorithm
4.4.4 Viscous dissipation term
4.4.5 Test case: higher-order semi-implicit viscosity solver
4.5 Taylor{Green Vortex
4.6 Taylor{Green vortex with uniform background ow
4.7 Taylor{Green vortex with viscosity
4.8 Conclusion & Future scope
5 Explicit Saint-Venant Schemes
5.1 Background
5.2 Basilisk O(2) Saint-Venant solver
5.2.1 First-order well balanced method
5.2.2 Second-order well balanced method
5.2.3 Riemann Solver
5.2.4 Time marching scheme { predictor-corrector algorithm
5.3 Test case – Linear surface gravity wave
5.3.1 WENO-based explicit Saint-Venant solver
5.4 Conclusion
6 Conclusion & Perspectives
