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Efficient numerical approximation of solutions to high-dimensional partial differential equations Universiteit Antwerpen
A lot of phenomena observed around us can be described in terms of mathematical problems or equations. Although the computational power to numerically solve these problems has extensively increased over the past decades, the mathematical equations to solve have become more and more complicated. The aim of this thesis is to study and develop efficient numerical methods to approximate solutions to high-dimensional partial differential equations ...
A difference scheme for multidimensional transfer equations with time delay Universiteit Gent
This paper continues research initiated in [29]. We develop a finite difference scheme for a first order multidimensional partial differential equation including a time delay. This class of equations is used to model different time lapse phenomena, e.g. birds migration, proliferation of viruses or bacteria and transfer of nuclear particles. For the constructed difference schemes the order of approximation, stability and convergence order are ...
The (Block) Macaulay Matrix: Solving Systems of Multivariate Polynomial Equations and Multiparameter Eigenvalue Problems KU Leuven
One of the most pervasive tools from (numerical) linear algebra is, without any doubt, the standard eigenvalue decomposition. Eigenvalues describe the intrinsic system dynamics of many natural and scientific phenomena; they explain how a system evolves along the eigenvector direction. This makes eigenvalues and eigenvectors indispensable in a wide array of problems. More importantly, at least in the context of this dissertation, the standard ...
Convergence of the Hundsdorfer-Verwer scheme for two-dimensional convection-diffusion equations with mixed derivative term Universiteit Antwerpen
Alternating Direction Implicit (ADI) schemes are popular in the numerical solution of multidimensional time-dependent partial differential equations (PDEs) arising in various contemporary application fields such as financial mathematics. The Hundsdorfer-Verwer (HV) scheme is an often used ADI scheme. A structural analysis of its fundamental properties, notably convergence, is of main interest. Up to now, however, a convergence result is only ...
Difference scheme for multidimensional transfer equation with time delay Universiteit Gent
Numerical magnetohydrodynamics KU Leuven
The ideal MagnetoHydroDynamic (MHD) equations accurately describe the macroscopic dynamics of a perfectly conducting plasma. Adopting a continuum, single fluid description in terms of the plasma density rho, velocity v, thermal pressure p and magnetic field B, the ideal MHD system expresses conservation of mass, momentum, energy, and magnetic flux. This nonlinear, conservative system of 8 partial differential equations enriches the Euler ...
Numerical magnetohydrodynamics KU Leuven
The ideal MagnetoHydroDynamic (MHD) equations accurately describe the macroscopic dynamics of a perfectly conducting plasma. Adopting a continuum, single fluid description in terms of the plasma density rho, velocity v, thermal pressure p and magnetic field B, the ideal MHD system expresses conservation of mass, momentum, energy, and magnetic flux. This nonlinear, conservative system of 8 partial differential equations enriches the Euler ...
Scalar hyperbolic PDE simulations and coupling strategies KU Leuven
We report on grid-adaptive, multi-dimensional simulations for hyperbolic PDEs, with a deliberate focus on the analytically tractable scalar case. Motivated by recent efforts towards multi-physics simulation strategies, we investigate a variety of coupling strategies for numerically solving hyperbolic partial differential equations (PDEs). We use adaptive mesh refinement in combination with shock-capturing spatio-temporal discretizations, and ...
Partial differential equation-based dense 3D structure and motion estimation from monocular image sequences Vrije Universiteit Brussel
In this study, the authors propose an approach towards dense depth reconstruction, combining robust feature-based structure from motion with the spatial coherence of dense reconstruction algorithms. To achieve this, a variational framework was set up, minimising the epipolar reprojection error and the image brightness constraint, while preserving discontinuities in the depth field by introducing an anisotropic diffusion term. As initial guess ...