Numerical Solution of Partial Differential Equations: Finite Difference MethodsSubstantially revised, this authoritative study covers the standard finite difference methods of parabolic, hyperbolic, and elliptic equations, and includes the concomitant theoretical work on consistency, stability, and convergence. The new edition includes revised and greatly expanded sections on stability based on the Lax-Richtmeyer definition, the application of Pade approximants to systems of ordinary differential equations for parabolic and hyperbolic equations, and a considerably improved presentation of iterative methods. A fast-paced introduction to numerical methods, this will be a useful volume for students of mathematics and engineering, and for postgraduates and professionals who need a clear, concise grounding in this discipline. |
Contents
FINITE | 11 |
CrankNicolson implicit method | 19 |
The stability of the elimination method | 27 |
27 | 46 |
ALTERNATIVE | 111 |
The local truncation errors associated with the Padé approxim | 124 |
The local truncation errors and symbols of extrapolation | 131 |
HYPERBOLIC EQUATIONS | 175 |
A worked example covering each method | 263 |
A sufficient condition for convergence | 269 |
Eigenvalues of the Jacobi and SOR iteration matrices and | 275 |
Theoretical determination of the optimum relaxation parame | 282 |
Introduction to 2cyclic matrices and consistent ordering | 288 |
The ordering vector for a block tridiagonal matrix | 294 |
Stones strongly implicit iterative method | 302 |
A recent direct method | 309 |
Common terms and phrases
2-cyclic A₁ analytical solution Assume aUlat ax² b₁ block tridiagonal boundary conditions calculate characteristic classical explicit coefficients components consistently ordered constant convergence corresponding Crank-Nicolson equations Crank-Nicolson method curve d₁ defined denote derivatives diagonal difference equations difference scheme discontinuity discretization error eigenvalues eigenvectors elimination equa equation au a²U equations approximating exact solution example Exercise extrapolation finite finite-difference equations finite-difference solution function U satisfies given gives Hence hyperbolic equation implicit initial conditions initial values iterative methods Jacobi iteration matrix Laplace's equation Lo-stable mesh lengths mesh points moduli non-zero numerical solution ordering vector ordinary differential equations Padé approximant parabolic equations partial differential equation Poisson's equation problem second-order shown shows sin² solution domain solution values solved spectral radius symmetric Table tends to zero theorem time-level tion tridiagonal matrix truncation error u₁ Ui,j Uj+1 unconditionally stable V₁ written