Numerical Methods for Partial Differential Equations

1 Numerical Methods for PartialDifferential EquationsSeongjai KimDepartment of Mathematics and StatisticsMississippi State UniversityMississippi State, MS 39762 USAE mail: 12, 2021 Seongjai Kim, Department of Mathematics and Statistics, Mississippi StateUniversity, Mississippi State, MS 39762-5921 USA Email: work of the author is supported in part by NSF grant the area of Numerical Methods for Differential Equations ", it seems veryhard to find a textbook incorporating mathematical, physical, and engineer-ing issues of Numerical Methods in a synergistic fashion. So the first goal ofthis lecture note is to provide students a convenient textbook that addressesboth physical and mathematical aspects of Numerical Methods for Partial dif-ferential Equations (PDEs).In solving PDEs numerically, the following are essential to consider: physical laws governing the Differential Equations (physical understand-ing), stability/accuracy analysis of Numerical Methods (mathematical under-standing), issues/difficulties in realistic applications, and implementation techniques (efficiency of human efforts).

2 In organizing the lecture note, I am indebted by Ferziger and Peric [23], John-son [32], Strikwerda [64], and Varga [68], among others. Currently the lecturenote is not fully grown up; other useful techniques would be soon questions, suggestions, comments will be deeply Mathematical Taylor s Theorem & Polynomial Fitting .. Finite Differences .. Uniformly spaced grids .. General grids .. Overview of PDEs .. Difference Equations .. Homework ..292 Numerical Methods for Taylor-Series Methods .. The Euler method .. Higher-order Taylor Methods .. Runge-Kutta Methods .. Second-order Runge-Kutta method .. Fourth-order Runge-Kutta method .. Adaptive Methods .. Accuracy Comparison for One-Step Methods .. Multi-step Methods .. High-Order Equations & Systems of Differential Equations .. Homework.

3 533 Properties of Numerical A Model Problem: Heat Conduction in 1D .. Consistency .. Convergence .. Stability .. Approaches for proving stability .. The von Neumann analysis .. Influence of lower-order terms .. Boundedness Maximum Principle .. Convection-dominated fluid flows .. Stability vs. boundedness .. Conservation .. A Central-Time Scheme .. The -Method .. Stability analysis for the -Method .. Accuracy order .. Maximum principle .. Error analysis .. Homework ..904 Finite Difference Methods for Elliptic Finite Difference (FD) Methods .. Constant-coefficient problems .. General diffusion coefficients .. FD schemes for mixed derivatives .. -norm error estimates for FD schemes .. The Algebraic System for FDM .. Solution of Linear Algebraic Systems .. Direct method: the LU factorization.

4 Linear iterative Methods .. Convergence theory .. Relaxation Methods .. Line relaxation Methods .. Krylov Subspace Methods .. Steepest descent method .. Conjugate gradient (CG) method .. Preconditioned CG method .. Other Iterative Methods .. Incomplete LU-factorization .. Numerical Examples with Python .. Homework .. 1505 Finite Element Methods for Elliptic Finite Element (FE) Methods in 1D Space .. Variational formulation .. Formulation of FEMs .. The Hilbert spaces .. An error estimate for FEM in 1D .. Other Variational Principles .. FEM for the Poisson equation .. Integration by parts .. Defining FEMs .. Assembly: Element stiffness matrices .. Extension to Neumann boundary conditions .. Finite Volume (FV) Method.

5 Average of The Diffusion Coefficient .. Abstract Variational Problem .. Numerical Examples with Python .. 2066 FD Methods for Hyperbolic Introduction .. Basic Difference Schemes .. Consistency .. Convergence .. Stability .. Accuracy .. Conservation Laws .. Euler Equations of gas dynamics .. Shocks and Rarefaction .. Characteristics .. Weak solutions .. Numerical Methods .. Modified Equations .. Conservative Methods .. Consistency .. Godunov s method .. Nonlinear Stability .. Total variation stability (TV-stability) .. Total variation diminishing (TVD) Methods .. Other nonoscillatory Methods .. Numerical Examples with Python .. Homework .. 2637 Domain Decomposition Introduction to DDMs.

6 Overlapping Schwarz Alternating Methods (SAMs) .. Variational formulation .. SAM with two subdomains .. Convergence analysis .. Coarse subspace correction .. Nonoverlapping DDMs .. Multi-domain formulation .. The Steklov-Poincar operator .. The Schur complement matrix .. Iterative DDMs Based on Transmission Conditions .. The Dirichlet-Neumann method .. The Neumann-Neumann method .. The Robin method .. Remarks on DDMs of transmission conditions .. Homework .. 294 Contentsvii8 Multigrid Methods Introduction to Multigrid Methods .. Homework .. 2999 Locally One-Dimensional Heat Conduction in 1D Space: Revisited .. Heat Equation in Two and Three Variables .. The -method .. Convergence analysis for -method .. LOD Methods for the Heat Equation.

7 The ADI method .. Accuracy of the ADI: Two examples .. The general fractional step (FS) procedure .. Improved accuracy for LOD procedures .. A convergence proof for the ADI-II .. Accuracy and efficiency of ADI-II .. Homework .. 33710 Special Propagation and Absorbing Boundary Conditions .. Introduction to wave Equations .. Absorbing boundary conditions (ABCs) .. Waveform ABC .. 34211 Projects FEMs for PDEs of One Spacial Variable .. 349A Basic Concepts in Fluid Conservation Principles .. Conservation of Mass .. Conservation of Momentum .. Non-dimensionalization of the Navier-Stokes Equations .. Generic Transport Equations .. Homework .. 359viiiContentsB Elliptic Partial Differential Regularity Estimates .. Maximum and Minimum Principles.

8 Discrete Maximum and Minimum Principles .. Coordinate Changes .. Cylindrical and Spherical Coordinates .. 368C Helmholtz Wave Equation 371D Richards s Equation for Unsaturated Water Flow 373E Orthogonal Polynomials and Orthogonal Polynomials .. Gauss-Type Quadratures .. 377F Some Mathematical Trigonometric Formulas .. Vector Identities .. 381G Finite Difference Formulas383 Chapter 1 Mathematical PreliminariesIn the approximation of derivatives, we consider the Taylor series expansionand the curve-fitting as two of most popular tools. This chapter begins witha brief review for these introductory techniques, followed by finite differenceschemes, and an overview of Partial Differential Equations (PDEs).In the study of Numerical Methods for PDEs, experiments such as the im-plementation and running of computational codes are necessary to under-stand the detailed properties/behaviors of the Numerical algorithm under con-sideration.

9 However, these tasks often take a long time so that the work canhardly be finished in a desired period of time. Particularly, it is the case forthe graduate students in classes of Numerical softwarewill beprovided to help you experience Numerical Methods 1. MATHEMATICAL Taylor s Theorem & Polynomial FittingWhile the Differential Equations are defined on continuous variables, their nu-merical solutions must be computed on a finite number of discrete points. Thederivatives should be approximated appropriately to simulate the physicalphenomena accurately and efficiently. Such approximations require variousmathematical and computational tools. In this section we present a brief re-view for the Taylor s series and the curve (Taylor s Theorem).Assume thatu Cn+1[a,b]and letc [a,b]. Then, for everyx (a,b), there is a point that lies betweenxandcsuch thatu(x) =pn(x) +En+1(x),( )wherepnis a polynomial of degree nandEn+1denotes the remainder definedaspn(x) =n k=0u(k)(c)k!

10 (x c)k, En+1(x) =u(n+1)( )(n+ 1)!(x c)n+ formula ( ) can be rewritten foru(x+h)(aboutx) as follows: forx, x+h (a,b),u(x+h) =n k=0u(k)(x)k!hk+u(n+1)( )(n+ 1)!hn+1( ) Taylor s Theorem & Polynomial Fitting3 Curve fittingAnother useful tool in Numerical analysis is thecurve fitting. It is often thecase that the solution must be represented as a continuous function ratherthan a collection of discrete values. For example, when the function is to beevaluated at a point which is not a grid point, the function must be interpo-lated near the point before the , we introduce the existence theorem for interpolating ,x1, ,xNbe a set of distinct points. Then, for arbi-trary real valuesy0,y1, ,yN, there is a unique polynomialpNof degree Nsuch thatpN(xi) =yi, i= 0,1, , 1. MATHEMATICAL PRELIMINARIESL agrange interpolating polynomialLet{a=x0< x1< < xN=b}be a partition of the interval[a,b].