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Fun and Games with Partial Differential Equations Craig Douglas CS 521 April 11, 2000 Definition of a PDE and Notation • A PDE is an equation with derivatives of at least two variables in it. • Let u be a function of x and y. There are several ways to write a PDE, e.g., ux uy u / x u / y • The equations above are linear and first order. The order is determined by the maximum number of derivatives of any term. • A nonlinear PDE has the solution times a partial derivative or a partial derivative raised to some power in it. Most interesting problems are nonlinear and time dependent. 2 Characterization of Simple Second Order PDE’s • Let auxx 2buxy cu yy dux eu y fu g • Then the type of PDE is determined by the discriminant b 2 ac – < 0 elliptic – = 0 parabolic – > 0 hyperbolic 3 Characterization of n Variable Second Order PDE’s • A general linear PDE of order 2: in, j 1 aij uxi x j in1 biuxi cu d . • Assume symmetry in coefficients so that A = [ aij ] is symmetric. Eig(A) are real. Let P and Z denote the number of positive and zero eigenvalues of A. – Elliptic: Z = 0 and P = n or Z = 0 and P = 0.. – Parabolic: Z > 0 (det(A) = 0). – Hyperbolic: Z=0 and P = 1 or Z = 0 and P = n-1. – Ultra hyperbolic: Z = 0 and 1 < P < n-1. 4 PDE Model Problems • Laplace’s Equation (elliptic): u xx u yy 0 • Heat Equation (parabolic): ut u xx u yy 0 • Wave Equations (hyperbolic): ut u x u y 0 utt u xx u yy 0 • All problems can be mapped to one of these!… in theory 5 Boundary and Initial Conditions, Well and Ill Posedness • Boundary conditions on G GD U GN U GR. – Dirichlet: u = g on GD. – Neumann: un = g on GN. – Robin: au + b un = g on GR. • Initial conditions at t=0. – U(t=0,x,y) = u0(x,y). • Well posed PDE if and only if – A solution to the problem exists. – The solution is unique. – The solution depends continuously on the problem data. • Ill posed if not well posed. 6 Example: Poisson Equation in 2D • – uxx – uyy = 1 in (0,1)2 ; u = 0 on (0,1)2 . 7 Finite Whosiwhatsit Methods • There are three common methods of producing a finite dimensional problem whose solution can be computed, which approximates the solution of the original, infinite dimensional problem: – Finite elements – Finite differences – Finite volumes • Each has its place, supporters, and detractors. • There are also other methods, e.g., collocation, spectral methods, pseudo-blah-blah-blah methods, etc. 8 Finite Differences • Assume we have a uniform mesh with a point x in the interior.. – Forward difference: D+h u(x) = u(x+h) – u(x). – Backward difference: D-h u(x) = u(x) – u(x-h). – Central difference: x u(x) = u(x+h/2) – u(x-h/2) or x2 u(x) = u(x+h) – 2u(x) + u(x-h). • Taylor Series and Truncation Error – Look at the difference between the approximation and the Taylor series. When they do not match, there is a remainder, which is known as the truncation error. It is usually specified as O(hp). 9 Poisson Equation Example, Again • The Poisson equation example used central differences to solve a block matrix problem of the form A = [-I, T, -I ], where I is the nxn identity matrix and T is a nxn tridiagonal matrix [ -1, 4, -1 ]. There are n rows of blocks in A (i.e., A is n2xn2). This is known as a 5 point operator. • Choosing the right finite element method on a square (right triangles with piecewise linear elements) leads to the same matrix problem. Choosing the elements differently can lead to a 9 point operator instead. 10 Finite Elements (Variational Formulations) • Find u in test space H such that a(u,v) = f(v) for all v in H, where a is a bilinear form and f is a linear functional. V ( x, y ) nj 1 V j j ( x, y ) I (V ) .5 i j AijViV j i biVi Aij Int (a deli . del j ) bi Int ( fi ) • The coefficients Vj are computed and the function V(x,y) is evaluated anyplace that a value is needed. • The basis functions should have local support (i.e., have a limited area where they are nonzero). 11 Matrix Free Methods • Many problems have simple matrices associated with the linear algebra (e.g., the Poisson equation example). • By using methods (e.g., Krylov space or relaxation methods) that only multiply the matrix A times a vector x, code to calculate y=Ax can be written instead of storing the matrix A. • This reduces the cost of the computer (which is mostly memory chips) and allows for vastly larger simulations. 12 Time Stepping Methods • Standard methods are common: – Forward Euler (explicit) – Backward Euler (implicit) – Crank-Nicolson (implicit) • Variable length time stepping – Most common in Method of Lines (MOL) codes or Differential Algebraic Equation (DAE) solvers 13 Parallel Computation • Serious calculations today are mostly done on a parallel computer. • The domain is partitioned into subdomains that may or may not overlap slightly. • Goal is to calculate as many things in parallel as possible even if some things have to be calculated on several processors in order to avoid communication. • Communication is the Darth Vader of parallel computing. 14 Example: Original Mesh Consider solving a problem on the given grid. Assume that only half of the nodes fit on a processor. 15 Example: Mesh on Two Processors • Dividing into two connected subsets and renumber within the subdomains. • Communication occurs between neighbors that cross the processor boundary. • Ghost points (or overlap) can reduce communication sometimes at the expense of extra computation. • Computation is o(1/1000) communication per word. 16 Mesh Decomposition • Goals are to maximize interior while minimizing connections between subdomains. Critical parameter: minimize communication. • Such decomposition problems have been studied in load balancing for parallel computation. • Lots of choices: • METIS package from the University of Minnesota. • PARTI package from the University of Maryland • … 17 Benchmarking: Speedup • Speedup for 5 layer SEOM. Dashed lines for large Pacific simulation (3552 elements) and the solid lines are for the small Atlantic Basin simulation (792 elements). Both simulations use 7th order spectral expansion. 18 Benchmarking: Timing • Timings versus processors for 5 layer SEOM. Dashed lines for large Pacific simulation (3552 elements) and the solid lines are for the small Atlantic Basin simulation (792 elements). Both simulations use 7th order spectral expansion. 19