Computational Methods

Two Dimensional PDEs Problem #1: Alternating Direction Implicit (ADI) Method Research and write a brief explanation of the ADI Method and solve: Two Dimensional LaPlace Equation: Electric Potential Over a Flat Plate with Point Charge ! ! ! ?2u(x, y) = f (x, y) for -1 = x = 1, -1 = y = 1 boundary conditions: u(x,y) = 0 for all boundaries f (0.5, 0.5) = -1 f (-0.5,-0.5) = 1 elsewhere : f (x, y) = 0 Two Dimensional Temperature Diffusion: ! ! ! 10-4 ?2u(x, y, t ) ?x2 + ?2u(x, y, t ) ?y2 ? ? ? ? ? ? = ?u(x, y, t ) ?t for 0 = x = 4, 0 = y = 4 0 = t = 5000 u(x, y,0) = 0 u(x, y, t ) = ey cos x - ex cos y for x = 0, x = 4, y = 0, y = 4 ! ! Present results for t= 5000 Problem #2: Crank-Nicolson Problem Solve the Two-Dimensional Temperature Problem above using Crank-Nicolson Method. Elliptic PDE ENGR516 Assignment #5: Two Dimensional PDEs Problem #1: Alternating Direction Implicit (ADI) Method Research and write a brief explanation of the ADI Method and solve: Two Dimensional LaPlace Equation: Electric Potential Over a Flat Plate with Point Charge ! ! ! ?2u(x, y) = f (x, y) for -1 = x = 1, -1 = y = 1 boundary conditions: u(x,y) = 0 for all boundaries f (0.5, 0.5) = -1 f (-0.5,-0.5) = 1 elsewhere : f (x, y) = 0 Two Dimensional Temperature Diffusion: ! ! ! 10-4 ?2u(x, y, t ) ?x2 + ?2u(x, y, t ) ?y2 ? ? ? ? ? ? = ?u(x, y, t ) ?t for 0 = x = 4, 0 = y = 4 0 = t = 5000 u(x, y,0) = 0 u(x, y, t ) = ey cos x - ex cos y for x = 0, x = 4, y = 0, y = 4 ! ! Present results for t= 5000 Problem #2: Crank-Nicolson Problem Solve the Two-Dimensional Temperature Problem above using Crank-Nicolson Method. Assignment #8 Expand the explicit method for hyperbolic equation to two dimensions and solve: Two Dimensional Wave Vibration Over Square Membrane ! ! ! 0.25 ?2u(x, y,t ) ?x2 + ?2u(x, y,t ) ?y2 ? ? ? ? ? ? = ?2u(x, y,t ) ?t 2 for 0 = x = 2, 0 = y = 2 and 0 = t = 2 u(0, y,t ) = 0, u(2, y,t ) = 0 u(x,0,t ) = 0, u(x,2,t ) = 0 u(x, y,0) = 0.1sin(p x)sin(p y / 2), ?u(x, y,0) ?t = 0 ! ! Present Results for t = 0.1 and t = 1.8