Introduction to discretisation methods

The objective of the course is to provide theoretical and practical competences in discretisation methods for solving differential equations of fluid and solid-state mechanics, and electromagnetic fields. The students acquire competences required for selecting and using a suitable discretisation method and analyzing the errors of the numerical solution.

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• Finite difference method (FDM)
• Finite volume method (FVM)
• Nonlinear high-order schemes in FDM and FVM
• Finite element method
• Boundary element method
• Methods for the solution of linear systems
• Verification and validation
• The assessment of the error of numerical solutions: convergence criteria for iterative methods, discretisation error, order of acuracy of a discrete numerical solution

Project: application of a discretisation method on a research problem

Knowledge and understanding:

The students acquire a knowledge of the theoretical fundamentals of discretisation methods for the solution of diferential equations. The following methods are presented: finite difference, finite volume, finite element, and boundary element methods. The students learn to use these numerical methods: from the formulation of the discretised differential equation in the form of a system of linear equations, over the numerical solution of the linear system, the application of convergence criteria for iterative methods, to the determination of the accuracy of the solution. They learn to quantify discretization errors, assess the convergence and determine the order of accuracy of the numerical solution. In a practical project they apply a selected discretision method on a research problem. Thereby they acquire skills in advanced application of the selected discretision method. They become competent in selecting and using a suitable discretisation method and analyzing the errors of the numerical solution.

• R. J. LeVeque, Finite difference methods for ordinary and partial differential equations, SIAM (Society for Industral and Applied Mathematics), 2007 E-version
• J.H. Ferziger, M. Perić, Computational Methods for Fluid Dynamics, 3rd edition, Springer, 2002 E-version
• J.Y. Murthy, Numerical Methods in Heat, Mass, and Momentum Transfer, Notes on Higher-Order Schemes, Purdue University, 2002 E-version
• O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method, 7th edition, Butterworth-Heinemann, 2013 1 Catalogue 2 Catalogue 3 Catalogue E-version
• L.C. Wrobel, M.H. Aliabadi, The Boundary Element Method, Wiley, 2002 1 Catalogue 2 Catalogue
• A.J. Nowak (Ed.). Numerical methods in heat transfer, (International studies in science and engineering, 11). Gliwice: Institute of Thermal Technology, Silesian University of Technology; Clausthal-Zellerfeld: Papierflieger, 2009 Catalogue
• B. A. Finlayson, The Method of Weighted Residuals and Variational Principles, Academic Press, 1972
• C.J. Roy, Review of code and solution verification procedures for computational simulation, Journal of Computational Physics 205 (2005), 131–156 https://doi.org/10.1016/j.jcp.2004.10.036 E-version
• R. Barrett et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, section 4.2: Stopping Criteria, SIAM (Soc. for Industral and Applied Mathematics), 1994.

Project demonstrating the application of a specific discretisation method to a specific engineering or scientific problem (50%). Oral exam (50%).

Associate professor of Material Science at the University of Nova Gorica.

Bibliography