The method of. fractional steps, known familiarly as the method oi splitting, is a remarkable technique, developed by N. N. Yanenko and his collaborators, for solving problems in theoretical mechanics numerically. It is applicable especially to potential problems, problems of elasticity and problems of fluid dynamics. Most of the applications at the present time have been to incompressible flow with free bound aries and to viscous flow at low speeds. The method offers a powerful means of solving the Navier-Stokes equations and the results produced so far cover a range of Reynolds numbers far…mehr
The method of. fractional steps, known familiarly as the method oi splitting, is a remarkable technique, developed by N. N. Yanenko and his collaborators, for solving problems in theoretical mechanics numerically. It is applicable especially to potential problems, problems of elasticity and problems of fluid dynamics. Most of the applications at the present time have been to incompressible flow with free bound aries and to viscous flow at low speeds. The method offers a powerful means of solving the Navier-Stokes equations and the results produced so far cover a range of Reynolds numbers far greater than that attained in earlier methods. Further development of the method should lead to complete numerical solutions of many of the boundary layer and wake problems which at present defy satisfactory treatment. As noted by the author very few applications of the method have yet been made to problems in solid mechanics and prospects for answers both in this field and other areas such as heat transfer are encouraging. As the method is perfected it is likely to supplant traditional relaxation methods and finite element methods, especially with the increase in capability of large scale computers. The literal translation was carried out by T. Cheron with financial support of the Northrop Corporation. The editing of the translation was undertaken in collaboration with N. N. Yanenko and it is a plea sure to acknowledge his patient help and advice in this project. The edited manuscript was typed, for the most part, by Mrs.
1. Uniform schemes.- 1.1 The class of problems under investigation. The Cauchy problem in Banach space.- 1.2 Uniform schemes.- 1.3 Examples.- 1.4 The method of factorization (sweep).- 1.5 The method of matrix factorization.- 2. Simple schemes in fractional steps for the integration of parabolic equations.- 2.1 The scheme of longitudinal-transverse sweep.- 2.2 The scheme of stabilizing corrections.- 2.3 The splitting scheme for the equation of heat conduction without a mixed derivative (orthogonal system of coordinates).- 2.4 The splitting scheme for the equation of heat conduction with a mixed derivative (arbitrary system of coordinates).- 2.5 The scheme of factorization of a difference operator.- 2.6 The scheme of approximate factorization of operators.- 2.7 The predictor-corrector scheme.- 2.8 Some remarks regarding schemes with fractional steps.- 2.9 Boundary conditions in the method of fractional steps for the heat conduction equation.- 3. Application of the method of fractional steps to hyperbolic equations.- 3.1 The simplest schemes for one-dimensional hyperbolic equations.- 3.2 Uniform implicit schemes for equations of hyperbolic type.- 3.3 Implicit schemes for hyperbolic equations in several dimensions.- 3.4 The splitting scheme of running computation.- 3.5 Method of approximate factorization for the wave equation...- 3.6 The method of splitting and majorant schemes.- 4. Application of the method of fractional steps to boundary value problems for Laplace's and Poisson's equations.- 4.1 The relation between steady and unsteady problems.- 4.2 The integration schemes of unsteady problems and iterative schemes.- 4.3 Iterative schemes for Laplace's equation in two dimensions S.- 4.4 Iterative schemes for Laplace's equation in three dimensions.- 4.5 Iterativeschemes for elliptic equations.- 4.6 Schemes with variable steps.- 4.7 Iterative schemes based on integration schemes for hyperbolic equations.- 4.8 Solution of the boundary value problem for Poisson's equation.- 4.9 Iterative schemes with averaging.- 4.10 Reduction of schemes of incomplete approximation to schemes of complete approximation.- 5. Boundary value problems in the theory of elasticity.- 5.1 The equation of elastic equilibrium and elastic vibrations.- 5.2 Boundary value problems in the theory of elasticity.- 5.3 The integration scheme for the unsteady equations of elasticity.- 5.4 Iterative schemes of solution of boundary value problems for the biharmonic equation.- 5.5 Iterative schemes for the system of equations of elastic displacements.- 5.6 Boundary conditions in problems of elasticity.- 6. Schemes of higher accuracy.- 6.1 Uniform schemes of higher accuracy.- 6.2 Factorized schemes of higher accuracy for the equation of heat conduction.- 6.3 Solution of Dirichlet's problem with the use of the schemes of higher accuracy.- 7. Integro-differential, integral, and algebraic equations.- 7.1 Equations of kinetics.- 7.2 Algebraic equations.- 8. Some problems of hydrodynamics.- 8.1 Potential flow past a contour.- 8.2 Potential flow of an incompressible heavy liquid with a free boundary (spillway problem).- 8.3 The flow of a viscous liquid.- 8.4 The method of channel flows.- 8.5 The predictor-corrector method (method of correctors).- 8.6 The equations of meteorology.- 9. General definitions.- 9.1 General formulation of the method of splitting. Validity of the method as determined by the elimination principle in the commutative case.- 9.2 Validity of the method of splitting in the non-commutative case.- 9.3 The method of approximate factorization of anoperator.- 9.4 The method of stabilizing corrections.- 9.5 The method of approximation corrections.- 9.6 The method of establishing the steady state.- 10. The method of weak approximation and the construction of the solution of the Cauchy problem in Banach space.- 10.1 Examples.- 10.2 A weak approximation for a system of differential equations.- 10.3 Convergence theorems.- References.
1. Uniform schemes.- 1.1 The class of problems under investigation. The Cauchy problem in Banach space.- 1.2 Uniform schemes.- 1.3 Examples.- 1.4 The method of factorization (sweep).- 1.5 The method of matrix factorization.- 2. Simple schemes in fractional steps for the integration of parabolic equations.- 2.1 The scheme of longitudinal-transverse sweep.- 2.2 The scheme of stabilizing corrections.- 2.3 The splitting scheme for the equation of heat conduction without a mixed derivative (orthogonal system of coordinates).- 2.4 The splitting scheme for the equation of heat conduction with a mixed derivative (arbitrary system of coordinates).- 2.5 The scheme of factorization of a difference operator.- 2.6 The scheme of approximate factorization of operators.- 2.7 The predictor-corrector scheme.- 2.8 Some remarks regarding schemes with fractional steps.- 2.9 Boundary conditions in the method of fractional steps for the heat conduction equation.- 3. Application of the method of fractional steps to hyperbolic equations.- 3.1 The simplest schemes for one-dimensional hyperbolic equations.- 3.2 Uniform implicit schemes for equations of hyperbolic type.- 3.3 Implicit schemes for hyperbolic equations in several dimensions.- 3.4 The splitting scheme of running computation.- 3.5 Method of approximate factorization for the wave equation...- 3.6 The method of splitting and majorant schemes.- 4. Application of the method of fractional steps to boundary value problems for Laplace's and Poisson's equations.- 4.1 The relation between steady and unsteady problems.- 4.2 The integration schemes of unsteady problems and iterative schemes.- 4.3 Iterative schemes for Laplace's equation in two dimensions S.- 4.4 Iterative schemes for Laplace's equation in three dimensions.- 4.5 Iterativeschemes for elliptic equations.- 4.6 Schemes with variable steps.- 4.7 Iterative schemes based on integration schemes for hyperbolic equations.- 4.8 Solution of the boundary value problem for Poisson's equation.- 4.9 Iterative schemes with averaging.- 4.10 Reduction of schemes of incomplete approximation to schemes of complete approximation.- 5. Boundary value problems in the theory of elasticity.- 5.1 The equation of elastic equilibrium and elastic vibrations.- 5.2 Boundary value problems in the theory of elasticity.- 5.3 The integration scheme for the unsteady equations of elasticity.- 5.4 Iterative schemes of solution of boundary value problems for the biharmonic equation.- 5.5 Iterative schemes for the system of equations of elastic displacements.- 5.6 Boundary conditions in problems of elasticity.- 6. Schemes of higher accuracy.- 6.1 Uniform schemes of higher accuracy.- 6.2 Factorized schemes of higher accuracy for the equation of heat conduction.- 6.3 Solution of Dirichlet's problem with the use of the schemes of higher accuracy.- 7. Integro-differential, integral, and algebraic equations.- 7.1 Equations of kinetics.- 7.2 Algebraic equations.- 8. Some problems of hydrodynamics.- 8.1 Potential flow past a contour.- 8.2 Potential flow of an incompressible heavy liquid with a free boundary (spillway problem).- 8.3 The flow of a viscous liquid.- 8.4 The method of channel flows.- 8.5 The predictor-corrector method (method of correctors).- 8.6 The equations of meteorology.- 9. General definitions.- 9.1 General formulation of the method of splitting. Validity of the method as determined by the elimination principle in the commutative case.- 9.2 Validity of the method of splitting in the non-commutative case.- 9.3 The method of approximate factorization of anoperator.- 9.4 The method of stabilizing corrections.- 9.5 The method of approximation corrections.- 9.6 The method of establishing the steady state.- 10. The method of weak approximation and the construction of the solution of the Cauchy problem in Banach space.- 10.1 Examples.- 10.2 A weak approximation for a system of differential equations.- 10.3 Convergence theorems.- References.
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