... method a degree of robustness for problems with discontinuities Let us remind you once again that scaling of the variables is often crucial for successful integration ofdifferentialequations The ... Second-Order Conservative Equations Usually when you have a system of high-order differentialequations to solve it is best to reformulate them as a system of first-order equations, as discussed ... is a particular class ofequations that occurs quite frequently in practice where you can gain about a factor of two in efficiency by differencing the equations directly The equations are second-order...
... 16 Integration of Ordinary DifferentialEquations Note that for compatibility with bsstep the arrays y and d2y are of length 2n for a system of n second-order equations The values of y are stored ... vol 27, pp 505–535 16.6 Stiff Sets ofEquations As soon as one deals with more than one first-order differential equation, the possibility of a stiff set ofequations arises Stiffness occurs in ... from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software...
... 16 Integration of Ordinary DifferentialEquations For multivalue methods the basic data available to the integrator are the first few terms of the Taylor series expansion of the solution at the ... The second of the equations in (16.7.9) is 752 Chapter 16 Integration of Ordinary DifferentialEquations you suspect that your problem is suitable for this treatment, we recommend use of a canned ... 1971, Numerical Initial Value Problems in Ordinary DifferentialEquations (Englewood Cliffs, NJ: Prentice-Hall), Chapter [1] Shampine, L.F., and Gordon, M.K 1975, Computer Solutionof Ordinary Differential...
... What are the variables? • What equations are satisfied in the interior of the region of interest? • What equations are satisfied by points on the boundary of the region of interest? (Here Dirichlet ... conceptually, to the solutionof large numbers of simultaneous algebraic equations When such equations are nonlinear, they are usually solved by linearization and iteration; so without much loss of generality ... problem as being the solutionof special, large linear sets ofequations As an example, one which we will refer to in §§19.4–19.6 as our “model problem,” let us consider the solutionof equation (19.0.3)...
... coefficients of the difference equations are so slowly varying as to be considered constant in space and time In that case, the independent solutions, or eigenmodes, of the difference equations are all of ... j 838 Chapter 19 PartialDifferentialEquations stable unstable ∆t ∆t ∆x ∆x x or j (a) ( b) Figure 19.1.3 Courant condition for stability of a differencing scheme The solutionof a hyperbolic ... rewritten as 840 Chapter 19 PartialDifferentialEquations Other Varieties of Error ξ = e−ik∆x + i − v∆t ∆x sin k∆x (19.1.25) An arbitrary initial wave packet is a superposition of modes with different...
... subject of stiff equations, relevant both to ordinary differentialequations and also to partialdifferentialequations (Chapter 19) Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... 1973, Computational Methods in Ordinary DifferentialEquations (New York: Wiley) Lapidus, L., and Seinfeld, J 1971, NumericalSolutionof Ordinary DifferentialEquations (New York: Academic Press) ... 708 Chapter 16 Integration of Ordinary DifferentialEquations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright...
... sequence of steps in identical manner Prior behavior of a solution is not used in its propagation This is mathematically proper, since any point along the trajectory of an ordinary differential ... is the routine for carrying out one classical Runge-Kutta step on a set of n differentialequations You input the values of the independent variables, and you get out new values which are stepped ... from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software...
... 19 PartialDifferentialEquations The physical interpretation of the restriction (19.2.6) is that the maximum allowed timestep is, up to a numerical factor, the diffusion time across a cell of ... Ames, W.F 1977, Numerical Methods for PartialDifferential Equations, 2nd ed (New York: Academic Press), Chapter Goldberg, A., Schey, H.M., and Schwartz, J.L 1967, American Journal of Physics, vol ... amplitudes, so that the evolution of the larger-scale features of interest takes place superposed with a kind of “frozen in” (though fluctuating) background of small-scale stuff This answer gives...
... These will occupy us for the remainder of the chapter CITED REFERENCES AND FURTHER READING: Ames, W.F 1977, Numerical Methods for PartialDifferential Equations, 2nd ed (New York: Academic Press) ... PDEs When you increase the order of a differencing method to greater than the order of the original PDEs, you introduce spurious solutions to the difference equations This does not create a problem ... αx sin ky ∆)2 856 Chapter 19 PartialDifferentialEquations (19.3.13) Called the alternating-direction implicit method (ADI), this embodies the powerful concept of operator splitting or time...
... effectively fewer equations than unknowns In this case there can be either no solution, or else more than one solution vector x In the latter event, the solution space consists of a particular solution ... combination of (typically) N − M vectors (which are said to be in the nullspace of the matrix A) The task of finding the solution space of A involves • Singular value decomposition of a matrix ... written as the N ×N set ofequations (AT · A) · x = (AT · b) (2.0.4) where AT denotes the transpose of the matrix A Equations (2.0.4) are called the normal equationsof the linear least-squares...