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MATH 5620 NUMERICAL ANALYSIS IIMIDTERM REVIEW SHEETChapter 5 – Initial Value Problems• §5.1 Lipschitz condition, well-posedness, existence and uniquenesstheorem for first order IVP.• §5.2-5.3: Taylor series method, know how to derive them. Notion oforder, local truncation error and global truncation error (in general).How to find local truncation error for Taylor series method.• §5.4: Runge-Kutta methods. Advantages over Taylor series meth-ods. Know general form of second order Runge-Kutta method, andwhy the parameters are related (match Taylor series). You don’tneed to know all the RK2 methods by their names. Why are RKmethods of order higher than 4 less attractive than RK4?• §5.5: Runge-Kutta-Fehlberg method. You have to know the basicidea (using embedded methods). Please do not learn the method byheart or its coefficients. Need to understand the derivation of updatefor h.• §5.6 Multistep methods. Understand the relationship with integra-tion formulas. Use the undetermined coefficient method. Differ-ence between Adams-Bashforth and Adams-Moulton. What is apredictor-corrector method? How is it related to a fixed point iter-ation?• Analysis of linear multi-step methods. Know the linear functionalL and how it can be used to get the order or the local truncationerror of a method. You do not need to know the formulas for the djexactly, but you should know where they come from.• §5.7 Variable step-size multistep methods. You need to know the bigidea to obtain adaptivity. How does this method compare to Runge-Kutta-Fehlberg from the practical point of view? As for RKF youmay be asked to estimate the size h to get a desired precision.• §5.8 Extrapolation methods: Know the main idea (clever linear com-binations to get error cancellations) and how to apply it to an ex-ample.• §5.9 Higher order equations and systems. Know how to transform analgorithm for the scalar case to system case. Know how to transforma higher order equation into a system.• §5.10 Stability, only for linear multistep methods. Difference be-tween implicit/explicit. Notion of convergent method. Consistencyand stability notions and how they relate to convergence. Know how12 MATH 5620 NUMERICAL ANALYSIS II MIDTERM REVIEW SHEETto determine if a given linear multistep method is consistent, stableor convergent. (you don’t need to know the proof that convergenceimplies stability and consistency).• Difference equations. Know how to solve a difference equation whenroots of characteristic polynomial.• §5.11 Stiff differential equations. A-stability and how to determineif a linear multi-step method is A-stable. What is the region ofabsolute stability and what does it mean? (both are linked to howthe method fares with the simple problem y0= λy; y(0) = 1 with agiven time step h).Chapter 11 – Boundary Value Problems• Theorem giving sufficient condition for the problem to have a uniquesolution.• §11.1–11.2: Linear shooting method and Non-Linear Shooting method.You should know the basic ideas behind these two methods. With-out knowing the formulas by heart you should be able to to linearlycombine two solutions to a linear IVP to get the solution to a linearBVP. Also you should understand how non-linear shooting works(with Newton’s method)• §11.3–11.4: Finite Difference Methods. For linear BVP know howto construct the linear system. For Non-Linear BVP: know how tosetup the non-linear system (Jacobian etc. . . ) and Newton’s method.Chapter 12 – Numerical solution to partial differentialequationsNote: What we saw in class is significantly different from what is coveredin the class textbook, so please refer to the class notes.• Elliptic problems: (Laplace equation ∆u = f) Five point stencilfor Laplacian and how the discretization works. Know that the localtruncation error is O(h2).• Parabolic problems: (heat equation ut= κ∆u).– Method of lines discretization and how relationship betweenTrapezoidal rule and Crank-Nicholson.– Please do not remember the expressions for the eigenvalues ofthe discrete Laplacian. However you should be able to use theseexpressions to determine the region of stability of a discretiza-tion method: for example for Euler’s method you should be ableto get that k/h2< 1/2 from the expression of the eigenvaluesfor the discrete Laplacian.– Know the idea of LOD and ADI for multidimensional problems.What does one gain by using these “operator splitting” meth-ods?• Hyperbolic problemsMATH 5620 NUMERICAL ANALYSIS II MIDTERM REVIEW SHEET 3– You need to understand Leapfrog, upwind, Lax-Friedrich andLax-Wendroff method. It is not necessary to remember thesemethods and/or their derivations by heart.– For a given method you should be able to show where the eigen-values of the iteration matrix lie (given the eigenvalues of thediscrete Laplacian and those of the matrix representing the dif-ference operator) and deduce the CFL condition from the sta-bility conditions (i.e. the eigenvalues of the iteration matrixshould be inside the absolute stability region of the methodused for the time discretization).• Finite Elements (only for Laplace equation −∆u = f)– Know weak formulation (1D and 2D)– Ritz-Galerkin approximation, Galerkin orthogonality, basic er-ror estimate (with the energy norm), optimality of the Ritz-Galerkin approximation in the energy norm.– Know what the Sobolev norms k · kL2(Ω)and k · kHk(Ω)as wellas the semi-norm | · |Hk(Ω)mean. There will be no derivation oferror estimates (and you do not need to know any other errorestimate than the basic one) but you may be asked to piecetogether inequalities to obtain an error estimate.– Know what are the local and global interpolants.– Know how to assemble the stiffness matrix and right hand sidefor 1D P1 elements (2 chain rules + change of integration vari-able).Chapter 9 – EigenvaluesNote: What we saw in class is significantly different from what is coveredin the class textbook, so please refer to the class notes.• Know how to write pseudocode for the power method, symmetricpower method, inverse iteration and Rayleigh quotient iteration.• You should be able to say what are the advantages/disadvantages ofsuch methods from the computational point of view.• Know the idea behind Householder reflectors and how they can beused to reduce a matrix to Hessenberg (or tridiagonal) form. Youdo not need to know how to choose the Householder reflector that ismore


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