Monotone Numerical Methods for Nonlinear Systems and Second Order Partial Differential Equations
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Multigrid methods are numerical solvers for partial differential equations (PDEs) that systematically exploit the relationship between approximate solutions on multiple grids to arrive at a solution whose accuracy is consistent with the finest grid but for considerably less work. These methods converge in a small number of constant iterations independent of the grid size and hence, are often dramatically more efficient than others. In this thesis, we develop multigrid methods for three different classes of PDEs. In addition, we also develop discretization schemes for two model problems. First, we propose multigrid methods based on upwind interpolation and restriction techniques for computing the steady state solutions for systems of one and two-dimensional nonlinear hyperbolic conservation laws. We prove that the two-grid method is total variation diminishing and the multigrid methods are consistent and convergent for one-dimensional linear systems. Second, we propose a fully implicit, positive coefficient discretization that converges to the viscosity solution for a two-dimensional system of Hamilton-Jacobi-Bellman (HJB) PDEs resulting from dynamic Bertrand duopoly. Furthermore, we develop fast multigrid methods for solving the systems of discrete nonlinear HJB PDEs. The new multigrid methods are general and can be applied to other systems of HJB and HJB-Isaacs (HJBI) PDEs resulting from American options under regime switching and American options with unequal lending/borrowing rates and stock borrowing fees under regime switching, respectively. We provide a theoretical analysis for the smoother, restriction and interpolation operators of the multigrid methods. Finally, we develop a fully implicit, unconditionally monotone finite difference numerical scheme, that converges to the viscosity solution of the three-dimensional PDE to price European options under a two-factor stochastic volatility model. The presence of cross derivative terms in high dimensional PDEs makes the construction of monotone discretization schemes challenging. We develop a wide stencil discretization based on a local coordinate transformation to eliminate the cross derivative terms. But, wide stencil discretization is first order accurate and computationally expensive compared to the second order fixed stencil discretization. Therefore, we use a hybrid stencil in which fixed stencil is used as much as possible and a wide stencil when the fixed stencil discretization does not satisfy the positive coefficient condition. We also develop fast multigrid methods to solve the discrete linear system.
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