Introduction to numerical schemes

Transcription

1 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes

2 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.

3 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

4 Boundary conditions Dirichlet: u(t, 0) = a u(t, 1) = b Neumann: u x (t, 0) = a u x (t, 1) = b Mixed boundary conditions: u(t, 0) = a, u x (t, 1) = b. Periodic: u(t, 0 ) = u(t, 1 + )

5 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

6 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).

7 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-order derivative is called the forward finite-difference approximation and is denoted by D + t u n m.

8 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 ). D 0 xxu n m = D x D + x u n m.

9 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 = 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.

10 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.

11 Stability and Convergence 1. Convergence: the discrete solution converges to the continuous solution in certain norm. lim u U. t, x 0 2. Consistency: the discrete difference operator solution converges to the continuous solution for every bounded u. lim L(u) L t, x(u) = 0 t, x 0

12 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). Lax s equivalence theorem: Given a properly 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

13 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).

14 CFL 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.

15 Observation: Maximum principle implies stability of the scheme. Lemma (Maximum principle) If K t 1/2, then ( x) 2 min m u0 m um n max m u0 m.

16 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 in a future tutorial).

17 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 The CFL condition: + t ) ( y) 2 (un m,p+1 2um,p n + um,p 1) n. ( t K ( x) 2 + t ) ( y) If x = y, then the CFL condition is: K t ( x)

18 The 2D heat equation. Numerical example Figure: Application of the heat eq. on a gray-scale image.