Lecture Notes on Numerical Methods for Partial Differential Equations

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1 Lecture Notes on Numerical Methods for Partial Differential Equations Jinn-Liang Liu Institute of Computational and Modeling Science National Tsing Hua University, Taiwan Lecture D Poisson s Equation and Finite Difference Method(FDM) 07/4/8 D Poisson s Problem: Givenafunctionf(x) andtwoconstantsg D andg N,findthesolutionu(x)satisfying u = f(x), x (0,)=Ω:Openset u(0) = g D DirichletBoundaryCondition u () = g N NeumannBC (a) (b) (c) HereΩisthedomainoftheproblemand Ω istheboundaryofω,ie, Ω= {0,},Ω=Ω Ω This is a simplified mathematical model, an ordinary differential equation with Dirichlet and Neumann boundary conditions(or a boundary value problem) To study a given complex mathematical model, we usually simplify the problem and then construct an exact (true, analytical) solution to the simplified problem With the solution, we can study the main properties of the problem under investigation In general, the exact solution of a realisticmodelisimpossibletofindbyanalyticalmethod(byhand) Wethen resort to a computer to find an approximate solution for us For this, we need a variety of numerical methods that can be implemented (written in a computer programming language such as C++) on the computer Before applying the numerical methods to the real problem, we must firstly verify the methods with a simplified problem for which the exact solution is already known so that we can check whether our numerical methods are effective and efficient This course is meant to teach you standard numerical methods for

2 Figure : The exact solution graph of Question (c) ordinary or partial differential equations(odes) or(pdes)(part ) and for linear algebra (Part ) Question Look very closely where the variable x is defined(interior orboundary) (a)whatistheunknownin(a)? (b)iff(x)=,canyou find a solution(call an exact solution) of(a)? More solutions? Infinitely many solutions (general solution)? (c) If g D = 0 and g N = 0, how many solutions youget? Can you drawapictureforyour solution(s)? (d)if we change the conditions u(0) = 0 and u () = 0 to u (0) = 0 and u () = 0, howmanysolutionsyouget? (e)nowifyouaregivenf(x)=sinx+cosx + lnx 4 +e sinx,canyouuseyourhandtofindanexactsolutionfor(a)? Now another important question is: How do we find an approximate solution of () for any arbitrary f(x), g D, and g N? This is the main purposeofthiscoursetoteachyouhowtofindanapproximationsolutionof anodeorpdeproblem HereisthesimplestmethodforProblem()in Part Part : Numerical Method for PDEs Finite Difference Method(FDM): Step Domain Discretization(Mesh Generation)

3 Uniform Mesh (Partition): We partition (discretize) the domain Ω = [0,] inton subintervals (meshes orelements) with uniformmeshsize =h= N andn mesh(grid)points (nodes)x i, i =, N Hence, x i =0+(i ),x i+ =0+i,x i+ =0+(i )etc Step Central Difference Approximation The following is the definition of a derivative that you learn from Calculus u u(x i +) u(x i ) (x i ) 0 u(x i ) u(x i ) 0 0 u i+ U i+ u(x i+ u i U i ) u(x i ) (Forward) () (Backward) (3) (Central) (4) (Central Difference) (5) (6) Notethedifferencebetweenu i (exact)andu i (approximation),ie,u i = u(x i ) U i whereu i areunknownscalarsthatwearelookingfor u u u i+ (x i ) i 0 0 lim 0 u i+ u i lim 0 u i u i (7) (8) u i u i +u i+ () (9) Substituting this expression into(a), we have U i U i +U i+ () =f i =f(x i ), i=,,n (0) TheDirichletBC(b)u(0)=u(x )=g D impliesthat U =g D () 3

4 whereas the Neumann BC(c) can be approximated by g N = u ()=u (x N ) () u(x N ) u(x N ) 0 (3) U N U N (4) U N U N = g N =h g N (5) Combining(),(0), and(), we obtain the system of linear algebraic equations: A NxNU = b 0 0 (6) A NxN = (7) U U g D U 3 h f U = U i b = h f i (8) h f N hg N U N whereaiscalledann byn coefficientmatrix, x isann byunknown vector,and b isann byknownvector In linear algebra, we usually use the notation A x = b for A U = b Donotconfuse x withthegridpointsx i Step3 SolvingtheLinearSystem A x = b (A U = b) 4

5 Question Write in detail from Domain Discretization to the linear systema U = b withn =5fortheexample f(x)=,g D =0andg N =0 Homework Can you define a(polynomial of degree m) function, say U(x), such that the graph of U(x) passes through the points (x i,u i )? Use the same degree of polynomial to define another function, say u I (x), such that its graph passes through the points (x i,u i ) Draw a picture to tell the difference between u(x), u I (x), and U(x) We say that u I (x) is a piecewiselinearinterpolation ofu(x)atx,x,,x N ifm=andn =5 Part: NumericalMethodforLinearAlgebraA x = b GivenAand b,therearetwowaystosolvea x = b fortheunknown vector x () Direct Methods: Gaussian Elimination etc These methods are appropriate for small systems with(n < 0000) () Iterative Methods: Jacobi, Gauss-Seidel, SOR, Conjugate-Gradient, etc for very large systems WewillspendmuchofourclasshoursonPartforthiscourse 5