Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University
Numerical Methods for Partial Differential Equations Finite Difference Methods for Elliptic Equations Finite Difference Methods for Parabolic Equations Finite Difference Methods for Hyperbolic Equations Finite Element Methods for Elliptic Equations
Finite Difference Methods for Elliptic Equations 1 Introduction 2 A Finite Difference Method for a Model Problem 3 General Finite Difference Approximations 4 Stability and Error Analysis of Finite Difference Methods
Finite Difference Methods for Elliptic Equations Introduction The definitions of the elliptic equations The definitions of the elliptic equations — 2nd order A general second order linear elliptic partial differential equation with n independent variables has the following form: n n ∂ 2 ∂ � � ± L ( u ) � ± u = f , a ij + b i + c (1) ∂ x i ∂ x j ∂ x i i , j =1 i =1 with (the key point in the definition) n n � � ξ 2 i , α ( x ) > 0 , ∀ ξ ∈ R n \{ 0 } , ∀ x ∈ Ω . (2) a ij ( x ) ξ i ξ j ≥ α ( x ) i , j =1 i =1 Note that (2) says the matrix A = ( a ij ( x )) is positive definite. 4 / 39
Finite Difference Methods for Elliptic Equations Introduction The definitions of the elliptic equations The definitions of the elliptic equations — 2nd order L – the 2nd order linear elliptic partial differential operator; a ij , b i , c — coefficients, functions of x = ( x 1 , . . . , x n ); f — right hand side term, or source term, a function of x ; The operator L and the equation (1) are said to be uniformly elliptic, if x ∈ Ω α ( x ) = α > 0 . inf (3) n n ∀ ξ ∈ R n \ { 0 } , ∀ x ∈ Ω . � � ξ 2 a ij ( x ) ξ i ξ j ≥ α i , α > 0 , i , j =1 i =1 5 / 39
Finite Difference Methods for Elliptic Equations Introduction The definitions of the elliptic equations The definitions of the elliptic equations — 2nd order ∂ 2 For example, △ = � n i is a linear second order uniformly i =1 ∂ x 2 elliptic partial differential operator, since we have here a ii = 1 , ∀ i , a ij = 0 , ∀ i � = j , and the Poisson equation −△ u ( x ) = f ( x ) is a linear second order uniformly elliptic partial differential equation. 6 / 39
Finite Difference Methods for Elliptic Equations Introduction The definitions of the elliptic equations The definitions of the elliptic equations — 2 m -th order A general linear elliptic partial differential equations of order 2 m with n independent variables has the following form: 2 m n ∂ k � � u = f , ± L ( u ) � ± + a 0 (4) a i 1 ,..., i k ∂ x i 1 . . . ∂ x i k k =1 i 1 ,..., i k =1 with (the key point in the definition) n n � � ξ 2 m a i 1 ,..., i 2 m ( x ) ξ i 1 · · · ξ i 2 m ≥ α ( x ) , i i 1 ,..., i 2 m =1 i =1 ∀ ξ ∈ R n \ { 0 } , ∀ x ∈ Ω . α ( x ) > 0 , (5) Note that (5) says the 2 m order tensor A = ( a i 1 ,..., i 2 m ) is positive definite. 7 / 39
Finite Difference Methods for Elliptic Equations Introduction The definitions of the elliptic equations The definitions of the elliptic equations — 2 m -th order L – the 2 m -th order linear elliptic partial differential operator; a i 1 ,..., i k , a 0 — coefficients, functions of x = ( x 1 , . . . , x n ); f — right hand side term, or source term, a function of x ; The operator L and the equation (4) are said to be uniformly elliptic, if x ∈ Ω α ( x ) = α > 0 . inf (6) n n � � ξ 2 m a i 1 ,..., i 2 m ( x ) ξ i 1 · · · ξ i 2 m ≥ α , i i 1 ,..., i 2 m =1 i =1 ∀ ξ ∈ R n \ { 0 } , ∀ x ∈ Ω . α > 0 , 8 / 39
Finite Difference Methods for Elliptic Equations Introduction The definitions of the elliptic equations The definitions of the elliptic equations As a typical example, the 2 m -th order harmonic equation ( −△ ) m u = f is a linear 2 m -th order uniformly elliptic partial differential equation, and △ m is a linear 2 m -th order uniformly elliptic partial differential operator, since we have here a i 1 ,..., i 2 m ( x ) = 1 , if the indexes appear in pairs; a i 1 ,..., i 2 m ( x ) = 0 , otherwise . In particular, the biharmonic equation △ 2 u = f is a linear 4th order uniformly elliptic partial differential equation, and △ 2 is a linear 4-th order uniformly elliptic partial differential operator. 9 / 39
Finite Difference Methods for Elliptic Equations Introduction Steady state convection-diffusion problem — a model problem for elliptic partial differential equations Steady state convection-diffusion equation 1 x ∈ Ω ⊂ R n ; 2 v ( x ): the velocity of the fluid at x ; 3 u ( x ): the density of certain substance in the fluid at x ; 4 a ( x ) > 0: the diffusive coefficient; 5 f ( x ): the density of the source or sink of the substance. 6 J : diffusion flux (measured by amount of substance per unit area per unit time) 7 Fick’s law: J = − a ( x ) ∇ u ( x ). 10 / 39
Finite Difference Methods for Elliptic Equations Introduction Steady state convection-diffusion problem — a model problem for elliptic partial differential equations Steady state convection-diffusion equation For an arbitrary open subset ω ⊂ Ω with piecewise smooth boundary ∂ω , Fick’s law says the substance brought into ω by diffusion per unit time is given by � � J · ( − ν ( x )) ds = a ( x ) ∇ u ( x ) · ν ( x ) ds , ∂ω ∂ω while the substance brought into ω by the flow per unit time is � u ( x ) v ( x ) · ( − ν ( x )) ds ∂ω and the substance produced in ω by the source per unit time is � f ( x ) dx . ω 11 / 39
Finite Difference Methods for Elliptic Equations Introduction Steady state convection-diffusion problem — a model problem for elliptic partial differential equations Steady state convection-diffusion equation Therefore, the net change of the substance in ω per unit time is d � � u ( x ) dx = a ( x ) ∇ u ( x ) · ν ( x ) ds d t ω ∂ω � � − u ( x ) v ( x ) · ν ( x ) ds + f ( x ) dx . ∂ω ω d � By the steady state assumption, ω u ( x ) dx = 0, for arbitrary ω , d t and by the divergence theorem (or Green’s formula or Stokes formula), this leads to the steady state convection-diffusion equation in the integral form � {∇ · ( a ∇ u − u v ) + f } dx = 0 , ∀ ω ω 12 / 39
Finite Difference Methods for Elliptic Equations Introduction Steady state convection-diffusion problem — a model problem for elliptic partial differential equations Steady state convection-diffusion equation The term − [ a ( x ) ∇ u ( x ) − u ( x ) v ( x )] is named as the substance flux, since it represents the speed that the substance flows. Assume that ∇ · ( a ∇ u − u v ) + f is smooth, then, we obtain the steady state convection-diffusion equation in the differential form −∇ · ( a ( x ) ∇ u ( x ) − u v ) = f ( x ) , ∀ x ∈ Ω . In particular, if v = 0 and a = 1, we have the steady state diffusion equation −△ u = f . 13 / 39
Finite Difference Methods for Elliptic Equations Introduction Boundary conditions Boundary conditions for the elliptic equations For a complete steady state convection-diffusion problem, or problems of elliptic equations in general, we also need to impose proper boundary conditions. Three types of most commonly used boundary conditions: First type u = u D , ∀ x ∈ ∂ Ω; ∂ u Second type ∂ν = g , ∀ x ∈ ∂ Ω; ∂ u Third type ∂ν + α u = g , ∀ x ∈ ∂ Ω; where α ≥ 0, and α > 0 at least on some part of the boundary ( physical meaning: higher density produces bigger outward diffusion flux ). 14 / 39
Finite Difference Methods for Elliptic Equations Introduction Boundary conditions Boundary conditions for the steady state convection-diffusion equation 1st type boundary condition — Dirichlet boundary condition; 2nd type boundary condition — Neumann boundary condition; 3rd type boundary condition — Robin boundary condition; Mixed-type boundary conditions — different types of boundary conditions imposed on different parts of the boundary. 15 / 39
Finite Difference Methods for Elliptic Equations Introduction General framework of Finite Difference Methods General framework of Finite Difference Methods 1 Discretize the domain Ω by introducing a grid; 2 Discretize the function space by introducing grid functions; 3 Discretize the differential operators by properly defined difference operators; 4 Solve the discretized problem to get a finite difference solution; 5 Analyze the approximate properties of the finite difference solution. 16 / 39
Finite Difference Methods for Elliptic Equations A Finite Difference Method for a Model Problem A Model Problem Dirichlet boundary value problem of the Poisson equation � −△ u ( x ) = f ( x ) , ∀ x ∈ Ω , u ( x ) = u D ( x ) , ∀ x ∈ ∂ Ω , where Ω = (0 , 1) × (0 , 1) is a rectangular region. 17 / 39
Finite Difference Methods for Elliptic Equations A Finite Difference Method for a Model Problem Finite Difference Discretization of the Model Problem Discretize Ω by introducing a grid 1 Space (spatial) step sizes: △ x = △ y = h = 1 / N ; 2 Index set of the grid nodes: J = { ( i , j ) : ( x i , y j ) ∈ Ω } ; 3 Index set of grid nodes on the Dirichlet boundary: J D = { ( i , j ) : ( x i , y j ) ∈ ∂ Ω } ; 4 Index set of interior nodes: J Ω = J \ J D . For simplicity, both ( i , j ) and ( x i , y j ) are called grid nodes. 18 / 39
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