Numerical Differential Equation Analysis Package

Numerical prototypal derivative Equation Analysis PackageThe Numerical Differential Equation Analysis big money combines executeality for analyzing derivative equations using boner point diagrams, Gaussian quadrature, and Newton-Cotes quadrature.ButcherRunge-Kutta systems argon expedient for numeric exclusivelyy solution certain types of ordinary differential equations. Deriving high- collection Runge-Kutta enjoins is no easy task, however. in that respect be several reasons for this. The first difficulty is in finding the so-c everyed evidence conditions. These are nonlinear equations in the coefficients for the method that must be satisfied to make the flaw in the method of company O (hn) for some integer n where h is the step size. The second difficulty is in solving these equations. Besides being nonlinear, in that respect is generally no unique solution, and m whatsoever heuristics and simplifying assumptions are usually made. Finally, there is the problem o f combinatorial explosion. For a twelfth- evidence method there are 7813 state conditionsThis package performs the first task finding the order conditions that must be satisfied. The result is uttered in terms of unknown coefficients aij, bj, and ci. The s-stage Runge-Kutta method to advance from x to x+h is thenwhereSums of the elements in the actors lines of the matrix aij occur repeatedly in the conditions imposed on aij and bj. In recognition of this and as a notational convenience it is usual to introduce the coefficients ci and the definitionThis definition is referred to as the row-sum condition and is the first in a sequence of row-simplifying conditions.If aij=0 for all ij the method is explicit that is, each of the Yi (x+h) is defined in terms of previously computed time set. If the matrix aij is not strictly set out triangular, the method is covert and requires the solution of a (generally nonlinear) system of equations for each timestep. A diagonally implicit metho d has aij=0 for all iThere are several shipway to express the order conditions. If the act of stages s is specified as a positive integer, the order conditions are expressed in terms of sums of explicit terms. If the publication of stages is specified as a symbol, the order conditions ordain involve symbolic sums. If the turn of stages is not specified at all, the order conditions will be expressed in stage-independent tensor notation. In addition to the matrix a and the vectors b and c, this notation involves the vector e, which is composed of all ones. This notation has two distinct advantages it is independent of the emergence of stages s and it is independent of the particular Runge-Kutta method.For further details of the theory plan the references.ai,jthe coefficient of f(Yj(x)) in the formula for Yi(x) of the methodbjthe coefficient of f(Yj(x)) in the formula for Y(x) of the methodcia notational convenience for aijea notational convenience for the vector (1, 1, 1, )Notat ion employ by functions for Butcher.RungeKuttaOrderConditionsp,sgive a tip of the order conditions that any s-stage Runge-Kutta method of order p must satisfyButcherPrincipalErrorp,sgive a list of the order p+1 terms appearing in the Taylor series expansion of the error for an order-p, s-stage Runge-Kutta methodRungeKuttaOrderConditionsp, ButcherPrincipalErrorpgive the result in stage-independent tensor notationFunctions associated with the order conditions of Runge-Kutta methods.ButcherRowSum mold whether the row-sum conditions for the ci should be explicitly included in the list of order conditionsButcherSimplifyspecify whether to apply Butchers row and column simplifying assumptionsSome options for RungeKuttaOrderConditions.This gives the number of order conditions for each order up through order 10. Notice the combinatorial explosion.In2=Out2=This gives the order conditions that must be satisfied by any first-order, 3-stage Runge-Kutta method, explicitly including the row-sum conditions.In3=Out3=These are the order conditions that must be satisfied by any second-order, 3-stage Runge-Kutta method. hither the row-sum conditions are not included.In4=Out4=It should be noted that the sums involved on the left-hand sides of the order conditions will be left in symbolic form and not expanded if the number of stages is left as a symbolic argument. This will greatly simplify the results for high-order, many-stage methods. An even more than compact form results if you do not specify the number of stages at all and the answer is given in tensor form.These are the order conditions that must be satisfied by any second-order, s-stage method.In5=Out5=Replacing s by 3 gives the same result asRungeKuttaOrderConditions.In6=Out6=These are the order conditions that must be satisfied by any second-order method. This uses tensor notation. The vector e is a vector of ones whose space is the number of stages.In7=Out7=The tensor notation brush aside likewise be expanded to gi ve the conditions in full.In8=Out8=These are the principal error coefficients for any third-order method.In9=Out9=This is a bound on the local error of any third-order method in the limit as h approaches 0, normalized to eliminate the effects of the ODE.In10=Out10=Here are the order conditions that must be satisfied by any fourth-order, 1-stage Runge-Kutta method. Note that there is no affirmable way for these order conditions to be satisfied there need to be more stages (the second argument must be larger) for there to be sufficiently many unknowns to satisfy all of the conditions.In11=Out11=RungeKuttaMethodspecify the type of Runge-Kutta method for which order conditions are being soughtExplicita setting for the option RungeKuttaMethod specifying that the order conditions are to be for an explicit Runge-Kutta methodDiagonallyImplicita setting for the option RungeKuttaMethod specifying that the order conditions are to be for a diagonally implicit Runge-Kutta methodImplicita settin g for the option RungeKuttaMethod specifying that the order conditions are to be for an implicit Runge-Kutta method$RungeKuttaMethoda global variable whose value can be set to Explicit, DiagonallyImplicit, or ImplicitControlling the type of Runge-Kutta method in RungeKuttaOrderConditions and related functions.RungeKuttaOrderConditions and certain related functions have the option RungeKuttaMethod with default setting $RungeKuttaMethod. Normally you will want to determine the Runge-Kutta method being considered by setting $RungeKuttaMethod to one of Implicit, DiagonallyImplicit, and Explicit, except you can specify an option setting or even change the default for an individual function.These are the order conditions that must be satisfied by any second-order, 3-stage diagonally implicit Runge-Kutta method.In12=Out12=An alternative (but less efficient) way to get a diagonally implicit method is to force a to be lower triangular by replacing upper-triangular elements with 0.In13=Out13 =These are the order conditions that must be satisfied by any third-order, 2-stage explicit Runge-Kutta method. The contradiction in the order conditions indicates that no such method is possible, a result which holds for any explicit Runge-Kutta method when the number of stages is less than the order.In14=Out14=ButcherColumnConditionsp,sgive the column simplifying conditions up to and including order p for s stagesButcherRowConditionsp,sgive the row simplifying conditions up to and including order p for s stagesButcherQuadratureConditionsp,sgive the quadrature conditions up to and including order p for s stagesButcherColumnConditionsp, ButcherRowConditionsp, etc.give the result in stage-independent tensor notationMore functions associated with the order conditions of Runge-Kutta methods.Butcher showed that the number and complexity of the order conditions can be reduced considerably at high orders by the toleration of so-called simplifying assumptions. For example, this reduction can be accomplished by adopting sufficient row and column simplifying assumptions and quadrature-type order conditions. The option ButcherSimplify in RungeKuttaOrderConditions can be used to determine these automatically.These are the column simplifying conditions up to order 4.In15=Out15=These are the row simplifying conditions up to order 4.In16=Out16=These are the quadrature conditions up to order 4.In17=Out17=Trees are fundamental objects in Butchers formalism. They yield both the derivative in a power series expansion of a Runge-Kutta method and the related order constraint on the coefficients. This package provides a number of functions related to Butcher trees.fthe elementary symbol used in the representation of Butcher treesButcherTreespgive a list, partitioned by order, of the trees for any Runge-Kutta method of order pButcherTreeSimplifyp,,give the set of trees through order p that are not reduced by Butchers simplifying assumptions, assuming that the quadrature conditions through order p, the row simplifying conditions through order , and the column simplifying conditions through order all hold. The result is grouped by order, starting with the first nonvanishing treesButcherTreeCountpgive a list of the number of trees through order pButcherTreeQtreegive True if the tree or list of trees tree is valid functional syntax, and False otherwiseConstructing and enumerating Butcher trees.This gives the trees that are needed for any third-order method. The trees are represented in a functional form in terms of the elementary symbol f.In18=Out18=This tests the validity of the syntax of two trees. Butcher trees must be constructed using multiplication, exponentiation or application of the function f.In19=Out19=This evaluates the number of trees at each order through order 10. The result is equivalent to Out2 but the calculation is much more efficient since it does not actually involve constructing order conditions or trees.In20=Out20=The previous result can b e used to calculate the total number of trees required at each order through order10.In21=Out21=The number of constraints for a method using row and column simplifying assumptions depends upon the number of stages. ButcherTreeSimplify gives the Butcher trees that are not reduced assuming that these assumptions hold.This gives the additional trees that are necessary for a fourth-order method assuming that the quadrature conditions through order 4 and the row and column simplifying assumptions of order 1 hold. The result is a single tree of order 4 (which corresponds to a single fourth-order condition).In22=Out22=It is often useful to be able to visualize a tree or forest of trees graphically. For example, depicting trees yields insight, which can in turn be used to aid in the construction of Runge-Kutta methods.ButcherPlottreegive a plot of the tree treeButcherPlottree1,tree2,give an array of plots of the trees in the forest tree1, tree2,Drawing Butcher trees.ButcherPlotColumnsspecif y the number of columns in the GraphicsGrid plot of a list of treesButcherPlotLabelspecify a list of plot labels to be used to label the nodes of the plotButcherPlotNodeSizespecify a scaling cistron for the nodes of the trees in the plotButcherPlotRootSizespecify a scaling factor for the highlighting of the root of each tree in the plot a zero value does not highlight rootsOptions to ButcherPlot.This plots and labels the trees through order 4.In23=Out23=In addition to generating and drawing Butcher trees, many functions are provided for measuring and manipulating them. For a complete verbal description of the importance of these functions, see Butcher.ButcherHeighttreegive the height of the tree treeButcherWidthtreegive the width of the tree treeButcherOrdertreegive the order, or number of vertices, of the tree treeButcherAlphatreegive the number of slipway of labeling the vertices of the tree tree with a totally ordered set of labels such that if (m, n) is an edge, then mButcherB etatreegive the number of ways of labeling the tree tree with ButcherOrdertree-1 distinct labels such that the root is not labeled, but every other vertex is labeledButcherBetan,treegive the number of ways of labeling n of the vertices of the tree with n distinct labels such that every leaf is labeled and the root is not labeledButcherBetaBartreegive the number of ways of labeling the tree tree with ButcherOrdertree distinct labels such that every node, including the root, is labeledButcherBetaBarn,treegive the number of ways of labeling n of the vertices of the tree with n distinct labels such that every leaf is labeledButcherGammatreegive the density of the tree tree the reciprocal of the density is the right-hand side of the order condition imposed by treeButcherPhitree,sgive the angle of the tree tree the weight (tree) is the left-hand side of the order condition imposed by treeButcherPhitreegive (tree) using tensor notationButcherSigmatreegive the order of the symmetry group o f isomorphisms of the tree tree with itselfOther functions associated with Butcher trees.This gives the order of the tree ffff f2.In24=Out24=This gives the density of the tree ffff f2.In25=Out25=This gives the elementary weight function imposed by ffff f2 for an s-stage method.In26=Out26=The subscript notation is a formatting device and the subscripts are really just the indexed variable NumericalDifferentialEquationAnalysisPrivate$i.In27=Out27//FullForm=It is also possible to obtain solutions to the order conditions using Solve and related functions. Many issues related to the construction Runge-Kutta methods using this package can be found in Sofroniou. The obligate also contains details concerning algorithms used in Butcher.m and discusses applications.Gaussian QuadratureAs one of its methods, the Mathematica function NIntegrate uses a fairly sophisticated Gauss-Kronrod-based algorithm. The Gaussian quadrature functionality provided in Numerical Differential Equation Analysis al lows you to easily study some of the theory toilet ordinary Gaussian quadrature which is a little less sophisticated.The basic idea behind Gaussian quadrature is to approximate the value if an integral as a linear combination of values of the integrand evaluated at specific pointsSince there are 2n free parameters to be chosen (both the abscissas xi and the weights wi) and since both integration and the sum are linear operations, you can expect to be able to make the formula correct for all polynomials of degree less than about 2n. In addition to penetrative what the optimal abscissas and weights are, it is often desirable to know how large the error in the approximation will be. This package allows you to answer both of these questions.GaussianQuadratureWeightsn,a,bgive a list of the pairs (xi, wi) to machine precision for quadrature on the interval a to bGaussianQuadratureErrorn,f,a,bgive the error to machine precisionGaussianQuadratureWeightsn,a,b,precgive a list of the pairs (xi, wi) to precision precGaussianQuadratureErrorn,f,a,b,precgive the error to precision precFinding formulas for Gaussian quadrature.This gives the abscissas and weights for the five-point Gaussian quadrature formula on the interval (-3, 7).In2=Out2=Here is the error in that formula. Unfortunately it involves the tenth derivative of f at an unknown point so you dont really know what the error itself is.In3=Out3=You can see that the error decreases rapidly with the length of the interval.In4=Out4=Newton-CotesAs one of its methods, the Mathematica function NIntegrate uses a fairly sophisticated Gauss-Kronrod based algorithm. Other types of quadrature formulas exist, each with their own advantages. For example, Gaussian quadrature uses values of the integrand at oddly spaced abscissas. If you want to integrate a function presented in tabular form at equally spaced abscissas, it wont campaign very well. An alternative is to use Newton-Cotes quadrature.The basic idea behind Newton-Cote s quadrature is to approximate the value of an integral as a linear combination of values of the integrand evaluated at equally spaced pointsIn addition, there is the question of whether or not to include the end points in the sum. If they are included, the quadrature formula is referred to as a shut formula. If not, it is an open formula. If the formula is open there is some ambiguity as to where the first abscissa is to be placed. The open formulas given in this package have the first abscissa one half step from the lower end point.Since there are n free parameters to be chosen (the weights) and since both integration and the sum are linear operations, you can expect to be able to make the formula correct for all polynomials of degree less than about n. In addition to knowing what the weights are, it is often desi