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chap11_8up

# chap11_8up - Partial Differential Equations Numerical...

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Unformatted text preview: Partial Differential Equations Numerical Methods for PDEs Sparse Linear Systems Scientiﬁc Computing: An Introductory Survey Chapter 11 – Partial Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c ± 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientiﬁc Computing 1 / 105 Partial Differential Equations Numerical Methods for PDEs Sparse Linear Systems Outline 1 Partial Differential Equations 2 Numerical Methods for PDEs 3 Sparse Linear Systems Michael T. Heath Scientiﬁc Computing 2 / 105 Partial Differential Equations Numerical Methods for PDEs Sparse Linear Systems Partial Differential Equations Characteristics Classiﬁcation Partial Differential Equations Partial differential equations (PDEs) involve partial derivatives with respect to more than one independent variable Independent variables typically include one or more space dimensions and possibly time dimension as well More dimensions complicate problem formulation: we can have pure initial value problem, pure boundary value problem, or mixture of both Equation and boundary data may be deﬁned over irregular domain Michael T. Heath Scientiﬁc Computing 3 / 105 Partial Differential Equations Numerical Methods for PDEs Sparse Linear Systems Partial Differential Equations Characteristics Classiﬁcation Partial Differential Equations, continued For simplicity, we will deal only with single PDEs (as opposed to systems of several PDEs) with only two independent variables, either two space variables, denoted by x and y , or one space variable denoted by x and one time variable denoted by t Partial derivatives with respect to independent variables are denoted by subscripts, for example u t = ∂u/∂t u xy = ∂ 2 u/∂x∂y Michael T. Heath Scientiﬁc Computing 4 / 105 Partial Differential Equations Numerical Methods for PDEs Sparse Linear Systems Partial Differential Equations Characteristics Classiﬁcation Example: Advection Equation Advection equation u t =-c u x where c is nonzero constant Unique solution is determined by initial condition u (0 , x ) = u ( x ) ,-∞ < x < ∞ where u is given function deﬁned on R We seek solution u ( t, x ) for t ≥ and all x ∈ R From chain rule, solution is given by u ( t, x ) = u ( x-c t ) Solution is initial function u shifted by c t to right if c > , or to left if c < Michael T. Heath Scientiﬁc Computing 5 / 105 Partial Differential Equations Numerical Methods for PDEs Sparse Linear Systems Partial Differential Equations Characteristics Classiﬁcation Example, continued Typical solution of advection equation, with initial function “advected” (shifted) over time < interactive example > Michael T. Heath Scientiﬁc Computing 6 / 105 Partial Differential Equations Numerical Methods for PDEs Sparse Linear Systems Partial Differential Equations Characteristics Classiﬁcation Characteristics Characteristics for PDE are level curves of solution...
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