Syllabus
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1. Accuracy and stability of basic numerical algorithms Numerical mathematics - accuracy of operations, computation errors, algorithm stability. Numerical integration and differentiation - integration with uniform and adaptive steps. Solution of ordinary differential equations - Euler, Runge-Kutta, and predictor-corrector methods. *
2. Linear algebra Second difference matrix, its eigenvalues and eigenvectors. Condition number of a matrix and its importance for numerical methods. *
3. Numerical solution of partial differential equations Finite difference method. Solution of boundary value problems - direct (Gauss elimination, LU decomposition, Fourier transform), indirect (relaxation methods - Jacobi, Gauss-Seidel...). Evolutionary equations, FTCS (forward time, centered space), Lax(-Friedrichs) method, Crank-Nicolson method. Von Neumann stability analysis, Courant Friedrichs Lewy condition. Principles of finite element method, weak formulation, discretization of the space of functions, practical demonstrations. *
4. Selected algorithms of computer physics Integral transforms - fast Fourier transform, deconvolution, Wiener and Lucy-Richardson deconvolution. Tikhonov regularization.
Annotation
The lecture deals with the accuracy and stability of numerical algorithms. Theoretical analysis as well as practical examples are demonstrated.
Special attention is paid to the solution of partial differential equations.