FUNDAMENTALS OF COMPUTATIONAL FLUID DYNAMICS

By I. MATAS

CONTENTS INTRODUCTION MOTIVATION BACKGROUND PROBLEM SPECI?CATION AND GEOMETRY PREPARATION SELECTION OF GOVERNING EQUATIONS AND BOUNDARY CONDITIONS SELECTION OF GRIDDING STRATEGY AND NUMERICAL METHOD ASSESSMENT AND INTERPRETATION OF RESULTS OVERVIEW NOTATION CONSERVATION LAWS AND THE MODEL EQUATIONS CONSERVATION LAWS THE NAVIER-STOKES AND EULER EQUATIONS THE LINEAR CONVECTION EQUATION DI?ERENTIAL FORM SOLUTION IN WAVE SPACE THE DI?USION EQUATION LINEAR HYPERBOLIC SYSTEMS PROBLEMS FINITE-DIFFERENCE APPROXIMATIONS MESHES AND FINITE-DI?ERENCE NOTATION SPACE DERIVATIVE APPROXIMATIONS FINITE-DI?ERENCE OPERATORS POINT DI?ERENCE OPERATORS MATRIX DI?ERENCE OPERATORS PERIODIC MATRICES CIRCULANT MATRICES CONSTRUCTING DI?ERENCING SCHEMES OF ANY ORDER TAYLOR TABLES GENERALIZATION OF DI?ERENCE FORMULAS LAGRANGE AND HERMITE INTERPOLATION POLYNOMIALS PRACTICAL APPLICATION OF PAD´E FORMULAS OTHER HIGHER-ORDER SCHEMES FOURIER ERROR ANALYSIS APPLICATION TO A SPATIAL OPERATOR DI?ERENCE OPERATORS AT BOUNDARIES THE SEMI-DISCRETE APPROACH REDUCTION OF PDE’S TO ODE’S THE MODEL ODE’S THE GENERIC MATRIX FORM EXACT SOLUTIONS OF LINEAR ODE’S EIGENSYSTEMS OF SEMI-DISCRETE LINEAR FORMS SINGLE ODE’S OF FIRST- AND SECOND-ORDER COUPLED FIRST-ORDER ODE’S GENERAL SOLUTION OF COUPLED ODE’S WITH COMPLETE EIGENSYSTEMS REAL SPACE AND EIGENSPACE DE?NITION EIGENVALUE SPECTRUMS FOR MODEL ODE’S EIGENVECTORS OF THE MODEL EQUATIONS SOLUTIONS OF THE MODEL ODE’S THE REPRESENTATIVE EQUATION FINITE-VOLUME METHODS BASIC CONCEPTS MODEL EQUATIONS IN INTEGRAL FORM ONE-DIMENSIONAL EXAMPLES A SECOND-ORDER APPROXIMATION TO THE CONVECTION EQUATION A FOURTH-ORDER APPROXIMATION TO THE CONVECTION EQUATION A SECOND-ORDER APPROXIMATION TO THE DI?USION EQUATION A TWO-DIMENSIONAL EXAMPLE TIME-MARCHING METHODS FOR ODE’S CONVERTING TIME-MARCHING METHODS TO O?E’S SOLUTION OF LINEAR O?E’S WITH CONSTANT COE?CIENTS FIRST- AND SECOND-ORDER DI?ERENCE EQUATIONS SPECIAL CASES OF COUPLED FIRST-ORDER EQUATIONS SOLUTION OF THE REPRESENTATIVE O?E’S THE OPERATIONAL FORM AND ITS SOLUTION EXAMPLES OF SOLUTIONS TO TIME-MARCHING O?E’S THE ? ?? RELATION ESTABLISHING THE RELATION THE PRINCIPAL ?-ROOT SPURIOUS ?-ROOTS ONE-ROOT TIME-MARCHING METHODS ACCURACY MEASURES OF TIME-MARCHING METHODS LOCAL AND GLOBAL ERROR MEASURES LOCAL ACCURACY OF THE TRANSIENT SOLUTION (ER?, |?| , ER?) LOCAL ACCURACY OF THE PARTICULAR SOLUTION (ERµ) TIME ACCURACY FOR NONLINEAR APPLICATIONS GLOBAL ACCURACY LINEAR MULTISTEP METHODS THE GENERAL FORMULATION EXAMPLES TWO-STEP LINEAR MULTISTEP METHODS PREDICTOR-CORRECTOR METHODS RUNGE-KUTTA METHODS 6 10 IMPLEMENTATION OF IMPLICIT METHODS 6 10 1 APPLICATION TO SYSTEMS OF EQUATIONS 6 10 2 APPLICATION TO NONLINEAR EQUATIONS 6 10 3 LOCAL LINEARIZATION FOR SCALAR EQUATIONS 6 10 4 LOCAL LINEARIZATION FOR COUPLED SETS OF NONLINEAR EQUATIONS 6 11 PROBLEMS STABILITY OF LINEAR SYSTEMS DEPENDENCE ON THE EIGENSYSTEM INHERENT STABILITY OF ODE’S THE CRITERION COMPLETE EIGENSYSTEMS DEFECTIVE EIGENSYSTEMS NUMERICAL STABILITY OF O?E ’S TIME-SPACE STABILITY AND CONVERGENCE OF O?E’S NUMERICAL STABILITY CONCEPTS IN THE COMPLEX ?-PLANE ?-ROOT TRACES RELATIVE TO THE UNIT CIRCLE STABILITY FOR SMALL ?T NUMERICAL STABILITY CONCEPTS IN THE COMPLEX ?H PLANE STABILITY FOR LARGE H UNCONDITIONAL STABILITY, A-STABLE METHODS STABILITY CONTOURS IN THE COMPLEX ?H PLANE FOURIER STABILITY ANALYSIS THE BASIC PROCEDURE SOME EXAMPLES RELATION TO CIRCULANT MATRICES CONSISTENCY CHOICE OF TIME-MARCHING METHODS STI?NESS DE?NITION FOR ODE’S RELATION TO ?-EIGENVALUES DRIVING AND PARASITIC EIGENVALUES STI?NESS CLASSI?CATIONS RELATION OF STI?NESS TO SPACE MESH SIZE PRACTICAL CONSIDERATIONS FOR COMPARING METHODS COMPARING THE E?CIENCY OF EXPLICIT METHODS IMPOSED CONSTRAINTS AN EXAMPLE INVOLVING DI?USION AN EXAMPLE INVOLVING PERIODIC CONVECTION COPING WITH STI?NESS EXPLICIT METHODS IMPLICIT METHODS A PERSPECTIVE STEADY PROBLEMS RELAXATION METHODS FORMULATION OF THE MODEL PROBLEM PRECONDITIONING THE BASIC MATRIX THE MODEL EQUATIONS CLASSICAL RELAXATION THE DELTA FORM OF AN ITERATIVE SCHEME THE CONVERGED SOLUTION, THE RESIDUAL, AND THE ERROR THE CLASSICAL METHODS THE ODE APPROACH TO CLASSICAL RELAXATION THE ORDINARY DI?ERENTIAL EQUATION FORMULATION ODE FORM OF THE CLASSICAL METHODS EIGENSYSTEMS OF THE CLASSICAL METHODS THE POINT-JACOBI SYSTEM THE GAUSS-SEIDEL SYSTEM THE SOR SYSTEM NONSTATIONARY PROCESSES 10 MULTIGRID 10 1 MOTIVATION 10 1 2 PROPERTIES OF THE ITERATIVE METHOD 10 2 THE BASIC PROCESS 10 3 A TWO-GRID PROCESS 10 4 PROBLEMS 11 NUMERICAL DISSIPATION 11 1 ONE-SIDED FIRST-DERIVATIVE SPACE DI?ERENCING 11 2 THE MODI?ED PARTIAL DI?ERENTIAL EQUATION 11 3 THE LAX-WENDRO?METHOD 11 4 UPWIND SCHEMES 11 4 1 FLUX-VECTOR SPLITTING 11 4 2 FLUX-DI?ERENCE SPLITTING 11 5 ARTI?CIAL DISSIPATION 11 6 PROBLEMS 12 SPLIT AND FACTORED FORMS 12 1 THE CONCEPT 12 2 FACTORING PHYSICAL REPRESENTATIONS — TIME SPLITTING 12 3 FACTORING SPACE MATRIX OPERATORS IN 2–D 12 3 1 MESH INDEXING CONVENTION 12 3 2 DATA BASES AND SPACE VECTORS 12 3 3 DATA BASE PERMUTATIONS 12 3 4 SPACE SPLITTING AND FACTORING 12 4 SECOND-ORDER FACTORED IMPLICIT METHODS 12 5 IMPORTANCE OF FACTORED FORMS IN 2 AND 3 DIMENSIONS 12 6 THE DELTA FORM 12 7 PROBLEMS 13 LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS 13 1 THE REPRESENTATIVE EQUATION FOR CIRCULANT OPERATORS 13 2 EXAMPLE ANALYSIS OF CIRCULANT SYSTEMS 13 2 1 STABILITY COMPARISONS OF TIME-SPLIT METHODS 13 2 2 ANALYSIS OF A SECOND-ORDER TIME-SPLIT METHOD 13 3 THE REPRESENTATIVE EQUATION FOR SPACE-SPLIT OPERATORS 13 4 EXAMPLE ANALYSIS OF 2-D MODEL EQUATIONS 13 4 1 THE UNFACTORED IMPLICIT EULER METHOD 13 4 2 THE FACTORED NONDELTA FORM OF THE IMPLICIT EULER METHOD 13 4 3 THE FACTORED DELTA FORM OF THE IMPLICIT EULER METHOD 13 4 4 THE FACTORED DELTA FORM OF THE TRAPEZOIDAL METHOD 13 5 EXAMPLE ANALYSIS OF THE 3-D MODEL EQUATION 13 6 PROBLEMS