Learn numerical analysis for partial differential equations.
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This course concerns the numerical discretization of partial differential equations (PDEs), and the implementation and testing thereof in realistic exercises. Parabolic and hyperbolic equations encountered in mathematics, physics, and engineering are discretized with finite difference and finite volume methods. The focus lies on PDEs with a time and one spatial dimension. Accuracy and stability of the numerical discretizations are considered in theoretical analysis, and this analysis is applied in practical exercises from science and engineering. After successful completion of the course, students are able to start designing, implementing and testing discretizations of PDEs in their own field. Practical assignments and a written test need to be completed. Recent assignments concerned topics such as: a model of lava eruptions (geophysics application), linear and nonlinear shallow water equations (civil engineering application), chemical fronts of reacting species (chemistry), dynamics of bacteria and diffusion of cancer cells in the brain (medical application).
Formal course outline
Introduction and classification of PDEs in parabolic, hyperbolic, and elliptic equations. Parabolic equations: accuracy and stability of finite difference approximations; Fourier analysis, explicit method; implicit method, Thomas algorithm, three-level scheme; more general boundary conditions, conservation; two dimensions, ADI. Hyperbolic equations: finite difference methods, analytical properties and numerical discretization of conservation laws, upwind and Lax-Wendroff schemes; finite volume methods, Riemann problems, conservative methods, Godunov scheme.
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