Solving Differential Equations by Multistep Initial and Boundary Value Methods
The numerical approximation of solutions of differential equations has been, and continues to be, one of the principal concerns of numerical analysis and is an active area of research. The new generation of parallel computers have provoked a reconsideration of numerical methods. This book aims to generalize classical multistep methods for both initial and boundary value problems; to present a self-contained theory which embraces and generalizes the classical Dahlquist theory; to treat nonclassical problems, such as Hamiltonian problems and the mesh selection; and to select appropriate methods for a general purpose software capable of solving a wide range of problems efficiently, even on parallel computers.
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Linear Difference Equations with Constant Coefficients
Polynomials and Toeplitz Matrices
Generalized Backward Differentiation Formulae
B Answers to the Exercises
Boundary Value Problems
Mesh Selection Strategies
Parallel Implementation of B2VMs
Extensions and Applications to Special Problems
Functions of matrices
The Kronecker Product
1 Use of Kronecker Product for Solving Matrix Equations
A-stable additional methods algorithm asymptotically stable B2VMs behavior block boundary conditions boundary locus boundary value problems BVMs BVPs Chapter characteristic polynomial coefficients complex plane computed solution condition number Consequently constant continuous problem continuous solution continuously invertible convergence corresponding defined definition denote difference equation discrete problem discrete solution eigenvalues entries ETR2s example Exercise fact Figure formula function GAMs GBDF of order Hamiltonian imaginary axis initial conditions initial value problems inversive IVPs k-step Lemma linear system mesh midpoint method Moreover Neumann polynomial nonsingular numerical methods obtained parallel parameters permutation matrix perturbation polynomial p(z processors Proof regular Jordan curve root of unit round-off errors satisfy scalar Schur polynomial Section solution of problem solved stability region stepsize h stiff Suppose symmetric schemes symplectic T-matrices test equation Theorem Toeplitz Toeplitz matrix trapezoidal rule unit circumference unit disk unit modulus vector verifies zero