MA2213 Lecture 11 PDE

Topics Introduction p Poisson equation p Visualization of numerical results p Boundary conditions p. 453 Finite difference grid p MATLAB Program p One-dimensional heat equation p

Introduction “Many phenomena in sciences and engineering depend on more than one variable. For example, an unknown function of a real-world problem usually depends on both time t and the location of the point (x,y,z).” p. 451 Physical laws, including the conservation of energy, momentum and mass, the laws of electricity and magnetism, thermodynamics, and chemical kinetics, require that the partial derivatives of these functions satisfy certain (partial differential) equations.

Introduction Increasingly, PDE’s are used to model biological and social phenomena. The models include the “law of supply and demand” in economics that determines equilibrium prices of goods and services, the Black-Sholes equation for options prices in arbitrage-free financial markets, and laws that describe the evolution of population densities that are used in epidemiology, ecology, and population genetics.

Introduction Examples Wave equation Poisson equation Heat equation

Poisson Equation Boundary Conditions Letbe a planar domain, and denote its The boundary value problemboundary by is called a Dirichlet problem because the value of u is specified on the boundary. For simplicity we will assume that and therefore

Finite Difference Grid For the square domain we choose a grid of points example boundary points interior points

Finite Difference Approximation We approximate the PDE by For boundary points where at interior points

Finite Difference Approximation We approximate Where, at every boundary point, we define by the solutionsof

Finite Difference Equations We now choose to obtain equations for and equations for

Solution Using Gauss-Seidel for i = 2,…,n for j = 2,…,n for k = 1,…,maxit Compare this with the Jacobi method that you used for problems in slides 37,41 of Lecture 7. end % j loop end % i loop end % k loop

MATLAB Program n = 10; maxit = 60;h = 1/n; [X,Y] = meshgrid(0:h:1,0:h:1); u = zeros(n+1); f = -2*pi^2*sin(pi*X).*sin(pi*Y); figure(1); surf(X,Y,f); title([‘f = -source distribution’]) for k = 1:maxit for i = 2:n for j = 2:n u(i,j) = 0.25*(u(i-1,j)+u(i+1,j)+u(i,j+1)+u(i,j-1)-h^2*f(i,j)); end figure(2); surf(X,Y,u); title([‘computed solution’]) utrue = sin(pi*X).*sin(pi*Y); figure(3); surf(X,Y,utrue-u); title([‘error with ’ num2str(maxit) ‘ iterations’])

Source Distribution

Computed Solution with 60 iter Gauss-Seidel

Error Due Primarily to Discretization

One Dimensional Heat Equation We consider the initial value problem for a function The heat equation arises in the modeling of a number ofmodeling phenomena and is often used in financial mathematics in thefinancial mathematics modeling of options. The famous Black-Scholes option pricingoptionsBlack-Scholes model's differential equation can be transformed into the heatdifferential equation equation allowing relatively easy solutions.

One Dimensional Heat Equation The domain of the function initial value boundary value

Discretization We divide the interval [0,L] intoequal parts to obtain a grid with size (space) grid points and We also choose maximum time and divide the interval [0,T] intoequal parts to obtain a grid with sizeand (time) grid points This provides a two-dimensional grid with points

Discretization We first use approximations We define and will compute approximations

Discretization and combine them with the Heat PDE to obtain approximate equations We replaceby

Discretization to obtain linear equations that can be solved directly by

Matrix Representation Define sequences Then whereand the matrix in slide 38, L7 has eig.val. the eigenvalues satisfy

Stability The numerical roundoff error sequencesatisfies therefore the algorithm is stable if and only if the iteration matrixhas spectral radius < 1 Since this occurs for largewhen

MATLAB CODE L = 1; nx = 20; hx = L/nx; a = 1; ht = hx^2/3; % for stable % ht = hx^2; for unstable niter = 500; F = zeros(nx-1,1); F(nx-1) = ht/(hx^2); u(:,1) = zeros(nx-1,1); A = -diag(2*ones(nx-1,1)) + diag(ones(nx-2,1),1) + diag(ones(nx-2,1),-1); I = eye(nx-1); c = a*ht/(hx^2); M = I + c*A; for k = 1:niter u(:,k+1) = M*u(:,k) + F; end

Stable Solution of Heat Equation

Unstable Solution of Heat Equation