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teaching:methcalchim:partial_differential_equation [2016/12/20 14:32] (Version actuelle)
villersd créée
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 +====== Numerical solutions of PDE ======
 +<note warning>​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.
 +</​note>​
  
 +  * [[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