THESIS
2021
1 online resource (xix, 68 pages) : color illustrations
Abstract
We develop simple numerical methods for solving hyperbolic conservation laws
defined on curves or surfaces. Following an embedding approach originally developed for eikonal
equations on manifolds, we will replace the hyperbolic conservation law on surfaces with a
related partial differential equation (PDE) defined in a tubular neighborhood of the implicit
surface. Then we can apply the typical finite difference method on the underlying Cartesian
mesh. We first discuss a simple extension idea, which orthogonally extends the flux function from
the implicit surface. This approach gives a simple first-order scheme for the surface hyperbolic
conservation laws. To improve the order of the numerical method, we apply the standard TVD-RK3
WENO3 scheme and introduce a modification fact...[
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We develop simple numerical methods for solving hyperbolic conservation laws
defined on curves or surfaces. Following an embedding approach originally developed for eikonal
equations on manifolds, we will replace the hyperbolic conservation law on surfaces with a
related partial differential equation (PDE) defined in a tubular neighborhood of the implicit
surface. Then we can apply the typical finite difference method on the underlying Cartesian
mesh. We first discuss a simple extension idea, which orthogonally extends the flux function from
the implicit surface. This approach gives a simple first-order scheme for the surface hyperbolic
conservation laws. To improve the order of the numerical method, we apply the standard TVD-RK3
WENO3 scheme and introduce a modification factor to the PDE based on the local curvature of the
surface in the second part of this work. This extra factor aims to correct the speed of the flux
on each level surface, so the numerical solution in the tubular neighborhood at the later time will
be orthogonal to the implicit surface. We will give two- and three-dimension examples on linear
and nonlinear equations to demonstrate the accuracy of the
proposed schemes.
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