# Didier Villers, UMONS - wiki

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teaching:methcalchim:partial_differential_equation

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 — teaching:methcalchim:partial_differential_equation [2016/12/20 14:32] (Version actuelle)villersd créée 2016/12/20 14:32 villersd créée 2016/12/20 14:32 villersd créée Ligne 1: Ligne 1: + ====== Numerical solutions of PDE ====== + ​In general, the numerical solutions of partial differential equations are obtained by standard methods developped and implemented in specialized softwares : + * Finite element method + * Finite volume method + * Boundary element method + * Spectral method + * Multigrid methods + * ... + These methods often make use of variational principles. + ​ + * [[wp>​Numerical_partial_differential_equations|Numerical partial differential equations]] + + However, the [[wp>​Finite_difference_method|finite difference method]] can be more easliy applied to a lot of classical PDE. In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values. + + One of the most usual problems concern the chemical and thermal diffusion problem, in either steady and unsteady conditions, for which same equations apply. The [[wp>​Heat_equation|heat equation]] is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time : + + $$\frac{\partial T}{\partial t} = \kappa\left(\frac{\partial^2T}{\partial x^2}+\frac{\partial^2T}{\partial y^2}+\frac{\partial^2T}{\partial z^2}\right)$$ + + * Dimensional analysis, dimensionless numbers,... + * Space discretization + * second-order central finite differences,​ first and second derivatives + * Time discretization,​ Euler approximation,​ explicit and implicit schemes + * Laplace equation (elliptic partial differential equation) + * Heat equation (parabolic partial differential equation) + * Error propagation,​ explicit //vs// implicit schemes, semi-implicit methods,and their correspondant time complexity + + ===== Applications ===== + * Heat diffusion, in stationary, unsteady conditions, with heat source (or sink), and various geometry + * Chemical diffusion + * Heat and chemical diffusion in heterogeneous media + * ion exchange chromatography
teaching/methcalchim/partial_differential_equation.txt · Dernière modification: 2016/12/20 14:32 par villersd