Tutorial 2. Introduction to numerical schemes
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1 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes c 2012
2 Classifying PDEs Looking at the PDE Au xx + 2Bu xy + Cu yy + Du x + Eu y + Fu +.. = 0, and its discriminant, B 2 AC, the equation is said to be: Parabolic if B 2 AC = 0 - for example, the heat equation, u t = u xx Hyperbolic if B 2 AC > 0 - for example, the second order wave equation, u tt = u xx Elliptic if B 2 AC < 0 - for example, Laplace equation, u xx + u yy = 0
3 Classifying PDEs (Cont.) This can be generalized to second order equations in N dimensions via the positive definiteness of the second order operator involved: ( ) ( ) T A B ( ) u + {lower order terms} = 0 B C
4 Heat equation The simple parabolic PDE with the initial values u t = K 2 u 2 x u(0, x) = u 0 (x) and some boundary conditions is called the (one-dimensional) heat equation.
5 This equation describes the thermal energy transport in a 1D rod, where u(t, x) describes the temperature at a point x at time t and K denotes the thermal conductivity constant. In 2D it describes the effect of defocusing on an image in optics The solution is given by convolution of the initial data with a Gaussian kernel: u(x, t) = G(x, t) u 0 (x) = G( x, t)u 0 (x x)d x R
6 Boundary conditions Dirichlet: u(t, 0) = a u(t, 1) = b Neumann: u x (t, 0) = a u x (t, 1) = b Reflective - Neumann with a = b = 0. Mixed boundary conditions: Periodic: u(t, 0) = a, u x (t, 1) = b. u(t, 0 ) = u(t, 1 + )
7 Discretization of the heat equation Replace the continuous system of coordinates (t, x) by a discrete grid (n, m) = (n t, m x), and the continuous function u(t, x) by a discrete version u n m = u(n t, m x). Figure: The grid
8 Finite differences 1. Replace the first-order time derivative u t (t, x) by a forward finite difference in time D t + um n = un+1 m um n t u t (t). 2. Replace the second-order space derivative u xx (t, x) by a central difference in space D 0 xxu n m = un m+1 2un m + u n m 1 ( x) 2 u xx (x).
9 Finite differences motivation From the Taylor expansion, u(t + t) = u(t) + u t (t) t u tt(t)( t) we obtain u t (t) = = u(t + t) u(t) t u(t + t) u(t) t 1 2 u tt(t) t O( t). This first-order approximation of the first derivative is called the forward finite-difference approximation and is denoted by D + t u n m.
10 Finite differences Forward difference D t + um n = un+1 m um n t u t (t) + O( t). In the same manner, backward difference can be defined Dt um n = un m um n 1 t u t (t) + O( t). To approximate a second-order derivative, use the central difference approximation D 0 xxu n m = un m+1 2un m + u n m 1 ( x) 2 u xx (x) + O(( x) 2 ). Note: We can also write D 0 xxu n m = D x D + x u n m.
11 Finite differences Proving the order of convergence by writing a fourth order Taylor series and canceling out terms, u n m+1 2un m + u n m 1 ( x) 2 = u n m + (u x ) n m x (u xx) n m ( x) ! (u xxx) n m ( x) ! (u xxxx) n m ( x)4 2u n m +u n m (u x ) n m x (u xx) n m ( x)2 1 3! (u xxx) n m ( x) ! (u xxxx) n m ( x)4 ( x) 2 = 1 2 (u xx) n m ( x) ! (u xxxx) n m ( x) (u xx) n m ( x) ! (u xxxx) n m ( x)4 ( x) 2
12 Finite differences = u xx (u xxxx) n m ( x)2 = u xx + O ( ( x) 2)
13 The discrete heat equation The continuous equation u t (t, x) = Ku xx (t, x) is replaced by or u n+1 m um n t = K un m+1 2un m + u n m 1 ( x) 2 um n+1 = um n + t ( x) 2 K ( um+1 n 2um n + um 1 n ), This scheme is explicit (called also Euler scheme) and very simple to implement.
14 We denote the continuous solution by U = u(t, x) and the discrete solution by u = u n m. We denote the continuous PDE operator L = t K xx. We denote the discrete operator L t, x = D + t KD 0 xx.
15 Stability and Convergence 1. Convergence: the discrete solution converges to the continuous solution in some norm. lim u U. t, x 0 2. Consistency: the discrete difference operator converges to the continuous for every bounded u. lim L(u) L t, x(u) = 0 t, x 0
16 Stability 3. Numerical stability: noise from initial conditions, numerical errors, etc. is not amplified. The numerical scheme is u n+1 = N u n Formally, stability means N > 0 C(N) > 0 s.t. n N N n C(N). If C is a constant for all N, this is known as unconditional stability. Lax s equivalence theorem: Given a well-posed initial value problem and a finite-difference approximation to it that satisfies the consistency condition, stability is the necessary and sufficient condition for convergence. consistency: stability convergence
17 The CFL condition for the wave equation The ( Courant-Friedrichs-Lewy (CFL) condition) is a necessary condition for advection problems limiting the maximum time step allowed in the difference equations. Specifically, this condition states that the domain of dependence of the differential equation must be included in the domain of dependence of the difference equation. Note: This does not apply to equations where information flows instantly such as the heat equation but the term CFL condition is often still used to denote stability limitations on step size in these cases.
18 A condition for stability of the discrete heat equation Denote r = K t ( x) 2. The finite difference scheme is um n+1 = um n + r ( um+1 n 2um n + um 1 n ) = (1 2r)um n + rum+1 n + rum 1 n If 1 2r > 0, we can write: u n+1 m (1 2r) max m Then u n+1 m (maximum principle property) un m + r max m un m + r max m un m max m un m(1 2r + 2r) = max m un m.
19 Observation: Maximum principle implies stability of the scheme.
20 Initial data 5 Solution to 1 D heat equation dt= Solurion to 1 D heat eq 4 x dt= dt= Figure: Computation of the discrete heat equation with Neumann boundary conditions. (h = 0.1) Top left: Initial data. Top right: stable solution of the heat equation conditions t = Bottom left: stable solution of the heat equation for t = Bottom right: unstable solution ( t = 0.006).
21 The 2D heat equation. Numerical scheme. u t = K(u xx + u yy ) ( t um,p n+1 = um,p n + K ( x) 2 (un m+1,p 2um,p n + um 1,p)+ n + t ) ( y) 2 (un m,p+1 2um,p n + um,p 1) n. A stabilty condition suggested by Courant, Friedrichs and Lewy: ( t K ( x) 2 + t ) ( y) If x = y, then the condition is: K t ( x)
22 Stability condition by Fourier analysis. Another way to check the stability of a numerical scheme is by Fourier analysis. Looking again at the heat equation: u t = u xx using the linearity of the equation we assume a solution of the form u n m = λ n e iξm x, where for stability we would like λ < 1. For the usual explicit scheme, we obtain: um n = un m+1 2un m + um 1 n t ( x) 2 λ n (λ 1)e iξm x t = ( x) 2 λn (e iξ x + e iξ x 2)e iξm x u n+1 m λ = 1 + t ( x) 2 (eiξ x + e iξ x 2)
23 Stability condition by Fourier analysis. finally arriving at λ = 1 + t ( x) 2 (2cos(ξ x) 2)? 1. Again we see that the condition < 1 ( x) 2 2 guarantees stability. The same method extends to higher dimensions. In the case of a finite time interval T, we can settle for a weaker form of stability, by demanding merely t λ < 1 + KT. This condition is known as the von-neumann condition. The analysis is often referred to as von-neumann analysis.
24 The 2D heat equation. Numerical example Figure: Application of the heat eq. on a gray-scale image.
25 Implicit finite difference schemes for the heat equation u n+1 m um n t = K un+1 m+1 2un+1 m + um 1 n+1 ( x) 2 Main advantage of implicit schemes: are unconditionally stable (i.e. stable for all time-steps and thus we can take large step times). (I ta)u n+1 = U n. Requires solving a linear system of equations. The matrix (I ta) is tridiagonal and can to be inverted using the Thomas algorithm (we will describe it in detail later on). Note: Assuring stability does not prevent the accuracy from deteriorating..
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