1 introduction
2 parabolic equations in one space variable
2.1 introduction
2.2 a model problem
2.3 series approximation
2.4 an explicit scheme for the model problem
2.5 difference notation and truncation error
2.6 convergence of the explicit scheme
2.7 fourier analysis of the error
2.8 an implicit method
2.9 the thomas algorithm
2.10 the weighted average or 0.method
2.11 a maximum principle and convergence
2.12 a three.time.level scheme
2.13 more general boundary conditions
2.14 heat conservation properties
2.15 more general linear problems
2.16 polar co.ordinates
2.17 nonlinear problems
bibliographic notes
exercises
3 2.d and 3.d parabolic equations
3.1 the explicit method in a rectilinear box
3.2 an adi method in two dimensions
3.3 adi and lod methods in three dimensions
3.4 curved boundaries
3.5 application to general parabolic problems
bibliographic notes
exercises
4 hyperbolic equations in one space dimension
4.1 characteristics
4.2 the cfl condition
4.3 error analysis of the upwind scheme
4.4 fourier analysis of the upwind scheme
4.5 the lax.wendroff scheme
4.6 the lax.wendroff method for conservation laws
4.7 finite volume schemes
4.8 the box scheme
4.9 the leap.frog scheme
4.10 hamiltonian systems and symplectic integration schemes
4.11 comparison of phase and amplitude errors
4.12 boundary conditions and conservation properties
4.13 extensions to more space dimensions
bibliographic notes
exercises
5 consistency, convergence and stability
5.1 definition of the problems considered
5.2 the finite difference mesh and norms
5.3 finite difference approximations
5.4 consistency, order of accuracy and convergence
5.5 stability and the lax equivalence theorem
5.6 calculating stability conditions
5.7 practical (strict or strong) stability
5.8 modified equation analysis
5.9 conservation laws and the energy method of analysis
5.10 summary of the theory
bibliographic notes
exercises
6 linear second order elliptic equations in
two dimensions
6.1 a model problem
6.2 error analysis of the model problem
6.3 the general diffusion equation
6.4 boundary conditions on a curved boundary
6.5 error analysis using a maximum principle
6.6 asymptotic error estimates
6.7 variational formulation and the finite element method
6.8 convection.diffusion problems
6.9 an example
bibliographic notes
exercises
7 iterative solution of linear algebraic equations
7.1 basic iterative schemes in explicit form
7.2 matrix form of iteration methods and their convergence
7.3 fourier analysis of convergence
7.4 application to an example
7.5 extensions and related iterative methods
7.6 the multigrid method
7.7 the conjugate gradient method
7.8 a numerical example: comparisons
bibliographic notes
exercises
references
index
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