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Last update: G_M (16.05.2012)
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Last update: T_KMA (27.09.2012)
L. C. Evans: Partial Differential Equations, AMS 2010 K. W. Morton, D. F. Mayers: Numerical solution of partial differential equations, 2nd ed., Cambridge University Press, Cambridge, 2005 J. C. Strikwerda: Finite difference schemes and partial differential equations, 2nd ed., SIAM, Philadelphia, 2004 A. Quarteroni, A. Valli: Numerical Approximation of Partial Differential Equations, Springer, 2008.
Doporučená studijní literatura a studijní pomůcky O. John, J. Nečas: Rovnice matematické fyziky, SPN 1972 M. Feistauer: Diskrétní metody řešení diferenciálních rovnic. Skripta, SPN, Praha, l98l S. J. Farlow: PDE for Scientists and Engineers, Dover, 1993 F. Sauvigny: Partial Differential Equations 1, Foundations and Integral Representations, Springer, 2006 |
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Last update: doc. Mgr. Petr Knobloch, Dr., DSc. (16.06.2015)
Basic examples of PDE's and their numerical solution by the finite difference method. Cauchy problem for a quasilinear PDE of the first order, transport equation, characteristics.
Von Neumann stability analysis of numerical schemes for Cauchy problems. Numerical solution of transport equation: CFL condition, upwinding, maximum principle, truncation error and approximation error, dissipation and dispersion.
Real analytic functions, Cauchy-Kowalevska Theorem, characteristic surfaces, classification of semilinear PDE's of the second order, transformation to canonical form.
Heat equation (fundamental solution, Cauchy problem, problem in bounded domain), wave equation (fundamental solution, Cauchy problem, energy methods).
Numerical solution of the mixed problem for heat equation: implicit and explicit schemes, theta-scheme, Fourier error analysis, maximum principle and convergence.
Relation between consistence, convergence and stability: general scheme for equations of the first order in time, Lax equivalence theorem.
Elliptic equations of the second order: fundamental solution of Laplace equation, Green's representation formula, Dirichlet problem for Laplace equation, mean value theorems, maximum principles.
Numerical solution of elliptic equations of the second order: approximation of general diffusion equation, derivation of schemes in irregular nodes, maximum principle and convergence. |