Numerical Methods for Differential Equations

1 Numerical Methods for Differential Equations chapter 5: Partial Differential Equations elliptic and parabolic . Gustaf Soderlind . and Carmen Arevalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis of Differential Equations , by Arieh Iserles and Introduction to Mathematical Modelling with Differential Equations , by Lennart Edsberg c Gustaf Soderlind, Numerical Analysis, Mathematical Sciences, Lund University, 2008-09. Numerical Methods for Differential Equations p. 1/50. 1. Brief overview of PDE problems Classification: Three basic types, four prototype Equations Elliptic u = 0 + BC. Parabolic ut = u + BC & IC. Hyperbolic utt = u + BC & IC.

2 Ut + a(u)ux = 0 + BC & IC. Numerical Methods for Differential Equations p. 2/50. Classification of PDEs Linear PDE with two independent variables Auxx + 2 Buxy + Cuyy + L(ux , uy , u, x, y) = 0. with L linear in ux , uy , u. Study . A B. := det = AC B 2. B C. Elliptic > 0 Parabolic = 0 Hyperbolic < 0. Numerical Methods for Differential Equations p. 3/50. Standard PDEs. Prototypical problems >0 Elliptic PDE Laplace equation uxx + uyy = 0 A = C = 1; B = 0. =0 Parabolic PDE Diffusion equation ut = uxx A = 1; B = C = 0. <0 Hyperbolic PDEs Wave equation utt = uxx A = 1; B = 0; C = 1. Numerical Methods for Differential Equations p. 4/50. PDE method types FDM Finite difference Methods FEM Finite element Methods FVM Finite volume Methods BEM Boundary element Methods We will mostly study FDM to cover basic theory Industrial relevance: FEM.

3 Numerical Methods for Differential Equations p. 5/50. PDE Methods for elliptic problems Simple geometry FDM or Fourier Methods Complex geometry FEM. Special problems FVM or BEM. Large sparse systems Combine with iterative solvers such as multigrid Methods Numerical Methods for Differential Equations p. 6/50. PDE Methods for parabolic problems Simple geometry FDM or Fourier Methods Complex geometry FEM. Stiffness Always use A stable time-stepping Methods Need Newton-type solvers for large sparse systems Numerical Methods for Differential Equations p. 7/50. PDE Methods for hyperbolic problems FDM, FVM. Sometimes FEM. Shocks Solutions may be discontinuous example: sonic boom.

4 Turbulence Multiscale phenomena Hyperbolic problems have several complications and many highly specialized techniques are often needed Numerical Methods for Differential Equations p. 8/50. 2. Elliptic problems. FDM. 2 2 2. Laplacian = 2. + 2+ 2. x y z Laplace equation u = 0. with boundary conditions u = u0 (x, y, z) x, y, z . Poisson equation u = f with boundary conditions u = u0 (x, y, z) x, y, z . Other boundary conditions also of interest ( = bdry of ). Numerical Methods for Differential Equations p. 9/50. Elliptic problems. Some applications Equilibrium problems Structural analysis (strength of materials). Heat distribution Potential problems Potential flow (inviscid, subsonic flow).

5 Electromagnetics (fields, radiation). Eigenvalue problems Acoustics Microphysics Numerical Methods for Differential Equations p. 10/50. Poisson equation an elliptic model problem 2u 2u 2. + 2 = f (x, y). x y Computational domain = [0, 1] [0, 1] (unit square). Dirichlet conditions u(x, y) = 0 on boundary Uniform grid {xi , yj }N,M. i,j=1 with equidistant mesh widths x = 1/(N + 1) and y = 1/(M + 1). Discretization Finite differences with ui,j u(xi , yj ). ui 1,j 2ui,j + ui+1,j ui,j 1 2ui,j + ui,j+1. 2. + 2. = f (xi , yj ). x y Numerical Methods for Differential Equations p. 11/50. Equidistant mesh x = y ui 1,j + ui,j 1 4ui,j + ui,j+1 + ui+1,j 2. = f (xi , yj ). x Participating approximations and mesh points yj+1.

6 Yj yj 1. xi 1 xi xi+1. Numerical Methods for Differential Equations p. 12/50. Computational stencil for x = y ui 1,j + ui,j 1 4ui,j + ui,j+1 + ui+1,j 2. = f (xi , yj ). x 1 Five-point operator . 1 4 1. 1. Numerical Methods for Differential Equations p. 13/50. The FDM linear system of Equations Lexicographic ordering of unknowns partitioned system . T I 0 .. u ,1 f (x , y1 ). I T I u ,2 f (x , y2 ).. 1 I T I u ,3 f (x , y3 ) . = .. 2. x .. I .. 0 I T u ,N f (x , yN ). with Toeplitz matrix T = tridiag(1 4 1). The system is N 2 N 2 , hence large and very sparse Numerical Methods for Differential Equations p. 14/50. 3. Elliptic problems. FEM. Finite Element Method PDE Lu = 0.

7 P P. Ansatz u= ci i Lu = ci L i Requirement h i , Lui = 0 gives coefficients {ci }. FEM is a least squares approximation, fitting a linear combination of basis functions { i } to the Differential equation using orthogonality Simplest case Piecewise linear basis functions Numerical Methods for Differential Equations p. 15/50. Strong and weak forms Strong form u = f ; u = 0 on . Take v with v = 0 on . Z Z. u v = f v . Integrate by parts to get weak form Z Z. u v = f v . Numerical Methods for Differential Equations p. 16/50. Strong and weak forms. 1D case Recall integration by parts in 1D. Z 1 Z 1. 1. u v = [ u v]0 + u v . 0 0. or in terms of an inner product hu , vi = hu , v i Generalization to 2D, 3D uses vector calculus Numerical Methods for Differential Equations p.

8 17/50. Weak form of u = f Define inner product Z. hv, ui = vu d .. and energy norm (note scalar product!). Z. a(v, u) = v u d .. to get the weak form of u = f as a(v, u) = hv, f i Numerical Methods for Differential Equations p. 18/50. Galerkin method (Finite Element Method). 1. Basis functions { i }. P. 2. Approximate u = cj j P R R. 3. Determine cj from cj i j = f i The cj are determined by the linear system Kc = F. The matrix K is called stiffness matrix R. Stiffness matrix elements kij = i j = a( i , j ). R. Right-hand side Fi = i f = h i , f i Numerical Methods for Differential Equations p. 19/50. The FEM mesh. Domain triangulation Piecewise linear basis { j } require domain triangulation 1.

9 0. 0 1. Numerical Methods for Differential Equations p. 20/50. 4. Parabolic problems The prototypical equation is the Diffusion equation ut = u Nonlinear diffusion ut = div (k(u)gradu). Boundary and initial conditions are needed Numerical Methods for Differential Equations p. 21/50. Parabolic problems. Some applications Diffusive processes Heat conduction ut = d uxx Chemical reactions Reaction diffusion ut = d uxx + f (u). 1. Convection diffusion ut = ux + u Pe xx Irreversibility ut = u is not well-posed! Numerical Methods for Differential Equations p. 22/50. Diffusion a parabolic model problem Equation ut = uxx Initial values u(x, 0) = g(x). Boundary values u(0, t) = u(1, t) = 0.

10 Separation of variables u(x, t) := X(x)T (t) . T X . ut = X T , uxx = X T = =: . T X. t . T = Ce X = A sin x + B cos x Numerical Methods for Differential Equations p. 23/50. Parabolic model problem.. Boundary values X(0) = X(1) = 0 k = (k )2 , therefore (k )2 t Xk (x) = 2 sin k x Tk (t) = e Initial values P . Fourier expansion g(x) = 1 ck 2 sin k x . X . (k )2 t u(x, t) = 2 ck e sin k x k=1. Numerical Methods for Differential Equations p. 24/50. 5. Method of lines (MOL) discretization In ut = uxx , discretize 2 / x2 by ui 1 2ui + ui+1. uxx . x2. System of ODEs (semidiscretization) u = T x u . 2 1. 1 1 2 1.. u = .. u x2 .. 1 2. Numerical Methods for Differential Equations p.