17.5 Automated Allocation of Mesh Points 783 17.5 Automated Allocation of Mesh Points In relaxation problems,you have to choose values for the independent variable at the mesh points.This is called allocating the grid or mesh.The usual procedure is to pick a plausible set of values and,if it works,to be content.If it doesn't work,increasing the number of points usually cures the problem. If we know ahead of time where our solutions will be rapidly varying,we can put more grid points there and less elsewhere.Alternatively,we can solve the problem first on a uniform mesh and then examine the solution to see where we should add more points.We then repeat the solution with the improved grid.The object of the exercise is to allocate points in such a way as to represent the solution accurately. 8 It is also possible to automate the allocation of mesh points,so that it is done "dynamically"during the relaxation process.This powerful technique not only improves the accuracy of the relaxation method,but also (as we will see in the next section)allows internal singularities to be handled in quite a neat way.Here we learn how to accomplish the automatic allocation. We want to focus attention on the independent variable a,and consider two alternative reparametrizations of it.The first,we term g;this is just the coordinate corresponding to the 6 mesh points themselves,so that g 1 at k =1,g =2 at k =2,and so on.Between any two mesh points we have Ag =1.In the change of independent variable in the ODEs from x to q, dy (17.5.1) dx 二g becomes Press. dy二gdg dx (17.5.2) dg 9 In terms of g,equation (17.5.2)as an FDE might be written dx yk-Yk-1- +西g =0 (17.5.3) IENTIFIC or some related version.Note that dr/dg should accompany g.The transformation between and g depends only on the Jacobian da/dg.Its reciprocal dg/da is proportional to the density of mesh points. Now,given the function y(x),or its approximation at the current stage of relaxation, we are supposed to have some idea of how we want to specify the density of mesh points. For example,we might want dg/dr to be larger where y is changing rapidly,or near to the boundaries,or both.In fact,we can probably make up a formula for what we would like 10-621 dg/d to be proportional to.The problem is that we do not know the proportionality constant. Numerical That is,the formula that we might invent would not have the correct integral over the whole range of r so as to make g vary from 1 to M,according to its definition.To solve this problem 43106 we introduce a second reparametrization (g),where Q is a new independent variable.The Recipes relation between Q and g is taken to be linear,so that a mesh spacing formula for do/dr differs only in its unknown proportionality constant.A linear relation implies PQ North Software. =0 (17.5.4) dg2 or,expressed in the usual manner as coupled first-order equations. dQ=中 d的 =0 dg do (17.5.5) where is a new intermediate variable.We add these two equations to the set of ODEs being solved. Completing the prescription,we add a third ODE that is just our desired mesh-density function,namely (e)=d迎 (17.5.6) dx dg dx
17.5 Automated Allocation of Mesh Points 783 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). 17.5 Automated Allocation of Mesh Points In relaxation problems, you have to choose values for the independent variable at the mesh points. This is called allocating the grid or mesh. The usual procedure is to pick a plausible set of values and, if it works, to be content. If it doesn’t work, increasing the number of points usually cures the problem. If we know ahead of time where our solutions will be rapidly varying, we can put more grid points there and less elsewhere. Alternatively, we can solve the problem first on a uniform mesh and then examine the solution to see where we should add more points. We then repeat the solution with the improved grid. The object of the exercise is to allocate points in such a way as to represent the solution accurately. It is also possible to automate the allocation of mesh points, so that it is done “dynamically” during the relaxation process. This powerful technique not only improves the accuracy of the relaxation method, but also (as we will see in the next section) allows internal singularities to be handled in quite a neat way. Here we learn how to accomplish the automatic allocation. We want to focus attention on the independent variable x, and consider two alternative reparametrizations of it. The first, we term q; this is just the coordinate corresponding to the mesh points themselves, so that q = 1 at k = 1, q = 2 at k = 2, and so on. Between any two mesh points we have ∆q = 1. In the change of independent variable in the ODEs from x to q, dy dx = g (17.5.1) becomes dy dq = g dx dq (17.5.2) In terms of q, equation (17.5.2) as an FDE might be written yk − yk−1 − 1 2 g dx dq k + g dx dq k−1 =0 (17.5.3) or some related version. Note that dx/dq should accompany g. The transformation between x and q depends only on the Jacobian dx/dq. Its reciprocal dq/dx is proportional to the density of mesh points. Now, given the function y(x), or its approximation at the current stage of relaxation, we are supposed to have some idea of how we want to specify the density of mesh points. For example, we might want dq/dx to be larger where y is changing rapidly, or near to the boundaries, or both. In fact, we can probably make up a formula for what we would like dq/dx to be proportional to. The problem is that we do not know the proportionality constant. That is, the formula that we might invent would not have the correct integral over the whole range of x so as to make q vary from 1 to M, according to its definition. To solve this problem we introduce a second reparametrization Q(q), where Q is a new independent variable. The relation between Q and q is taken to be linear, so that a mesh spacing formula for dQ/dx differs only in its unknown proportionality constant. A linear relation implies d2Q dq2 =0 (17.5.4) or, expressed in the usual manner as coupled first-order equations, dQ(x) dq = ψ dψ dq =0 (17.5.5) where ψ is a new intermediate variable. We add these two equations to the set of ODEs being solved. Completing the prescription, we add a third ODE that is just our desired mesh-density function, namely φ(x) = dQ dx = dQ dq dq dx (17.5.6)
784 Chapter 17.Two Point Boundary Value Problems where o()is chosen by us.Written in terms of the mesh variable g,this equation is dx 西=石 (17.5.7) Notice that ()should be chosen to be positive definite,so that the density of mesh points is everywhere positive.Otherwise (17.5.7)can have a zero in its denominator. To use automated mesh spacing,you add the three ODEs (17.5.5)and (17.5.7)to your set of equations,i.e.,to the array y[j][k].Now r becomes a dependent variable!Q and also become new dependent variables.Normally,evaluating o requires little extra work since it will be composed from pieces of the g's that exist anyway.The automated procedure allows one to investigate quickly how the numerical results might be affected by various strategies for mesh spacing.(A special case occurs if the desired mesh spacing function Q can be found 81 analytically,i.e.,dQ/dr is directly integrable.Then,you need to add only two equations, those in 17.5.5,and two new variables . As an example of a typical strategy for implementing this scheme,consider a system with one dependent variable y(r).We could set ICAL dQ=+dm到 (17.5.8) △ 6 or )=9=1+/ RECIPES dx 4+ y6 (17.5.9) where A and 6 are constants that we choose.The first term would give a uniform spacing in if it alone were present.The second term forces more grid points to be used where y is changing rapidly.The constants act to make every logarithmic change in y of an amount o about as "attractive"to a grid point as a change in of amount A.You adjust the constants according to taste.Other strategies are possible,such as a logarithmic spacing in replacing de in the first term with dIn x. IENTIFIC CITED REFERENCES AND FURTHER READING: Eggleton,P.P.1971,Monthly Notices of the Royal Astronomical Society.vol.151,pp.351-364. 6 Kippenhan,R.,Weigert,A.,and Hofmeister,E.1968,in Methods in Computational Physics, vol.7 (New York:Academic Press),pp.129ff. 17.6 Handling Internal Boundary Conditions Numerical 色 or Singular Points 431086 Singularities can occur in the interiors of two point boundary value problems.Typically, (outside there is a point s at which a derivative must be evaluated by an expression of the form North Software. N(xs,y) S(xs)=Dxs习) (17.6.1) where the denominator D(s,y)=0.In physical problems with finite answers,singular points usually come with their own cure:Where D-0,there the physical solution y must be such as to make N-0 simultaneously,in such a way that the ratio takes on a meaningful value.This constraint on the solution y is often called a regularity condition.The condition that D(,y)satisfy some special constraint at is entirely analogous to an extra boundary condition,an algebraic relation among the dependent variables that must hold at a point. We discussed a related situation earlier,in $17.2,when we described the"fitting point method"to handle the task of integrating equations with singular behavior at the boundaries. In those problems you are unable to integrate from one side of the domain to the other
784 Chapter 17. Two Point Boundary Value Problems Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). where φ(x) is chosen by us. Written in terms of the mesh variable q, this equation is dx dq = ψ φ(x) (17.5.7) Notice that φ(x) should be chosen to be positive definite, so that the density of mesh points is everywhere positive. Otherwise (17.5.7) can have a zero in its denominator. To use automated mesh spacing, you add the three ODEs (17.5.5) and (17.5.7) to your set of equations, i.e., to the array y[j][k]. Now x becomes a dependent variable! Q and ψ also become new dependent variables. Normally, evaluating φ requires little extra work since it will be composed from pieces of the g’s that exist anyway. The automated procedure allows one to investigate quickly how the numerical results might be affected by various strategies for mesh spacing. (A special case occurs if the desired mesh spacing function Q can be found analytically, i.e., dQ/dx is directly integrable. Then, you need to add only two equations, those in 17.5.5, and two new variables x, ψ.) As an example of a typical strategy for implementing this scheme, consider a system with one dependent variable y(x). We could set dQ = dx ∆ + |d ln y| δ (17.5.8) or φ(x) = dQ dx = 1 ∆ + dy/dx yδ (17.5.9) where ∆ and δ are constants that we choose. The first term would give a uniform spacing in x if it alone were present. The second term forces more grid points to be used where y is changing rapidly. The constants act to make every logarithmic change in y of an amount δ about as “attractive” to a grid point as a change in x of amount ∆. You adjust the constants according to taste. Other strategies are possible, such as a logarithmic spacing in x, replacing dx in the first term with d ln x. CITED REFERENCES AND FURTHER READING: Eggleton, P. P. 1971, Monthly Notices of the Royal Astronomical Society, vol. 151, pp. 351–364. Kippenhan, R., Weigert, A., and Hofmeister, E. 1968, in Methods in Computational Physics, vol. 7 (New York: Academic Press), pp. 129ff. 17.6 Handling Internal Boundary Conditions or Singular Points Singularities can occur in the interiors of two point boundary value problems. Typically, there is a point xs at which a derivative must be evaluated by an expression of the form S(xs) = N(xs, y) D(xs, y) (17.6.1) where the denominator D(xs, y)=0. In physical problems with finite answers, singular points usually come with their own cure: Where D → 0, there the physical solution y must be such as to make N → 0 simultaneously, in such a way that the ratio takes on a meaningful value. This constraint on the solution y is often called a regularity condition. The condition that D(xs, y) satisfy some special constraint at xs is entirely analogous to an extra boundary condition, an algebraic relation among the dependent variables that must hold at a point. We discussed a related situation earlier, in §17.2, when we described the “fitting point method” to handle the task of integrating equations with singular behavior at the boundaries. In those problems you are unable to integrate from one side of the domain to the other