Transcription of 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).
2 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).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.
3 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 ..533 Properties of Numerical A Model Problem: Heat Conduction in 1D .. Consistency .. Convergence .. Stability.
4 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.
5 Solution of Linear Algebraic Systems .. Direct method: the LU factorization .. 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.
6 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 .. 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.
7 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 .. Overlapping Schwarz Alternating Methods (SAMs) .. Variational formulation .. SAM with two subdomains .. Convergence analysis.
8 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.
9 Convergence analysis for -method .. LOD Methods for the Heat Equation .. 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.
10 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 .. 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.
