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A Numerical Method of Characteristics for Solving Hyperbolic Partial Differential Equations David Lenz Simpson Iowa State University

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A Numerical Method of Characteristics for Solving Hyperbolic Partial Differential Equations David Lenz Simpson Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1967 A numerical method of characteristics for solving hyperbolic partial differential equations David Lenz Simpson Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Simpson, David Lenz, "A numerical method of characteristics for solving hyperbolic partial differential equations " (1967). Retrospective Theses and Dissertations. 3430. https://lib.dr.iastate.edu/rtd/3430 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. This dissertation has been microfilmed exactly as received 68-2862 SIMPSON, David Lenz, 1938- A NUMERICAL METHOD OF CHARACTERISTICS FOR SOLVING HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS. Iowa State University, PluD., 1967 Mathematics University Microfilms, Inc., Ann Arbor, Michigan A NUMERICAL METHOD OF CHARACTERISTICS FOR SOLVING HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS by David Lenz Simpson A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Mathematics Approved: Signature was redacted for privacy. In Cijarge of Major Work Signature was redacted for privacy. Headead of^ajorof Major DepartmentDepartmen Signature was redacted for privacy. Dean Graduate Co11^^ Iowa State University of Science and Technology Ames, Iowa 1967 ii TABLE OF CONTENTS Page I. INTRODUCTION 1 II. DISCUSSION OF THE ALGORITHM 3 III. DISCUSSION OF LOCAL ERROR, EXISTENCE AND UNIQUENESS THEOREMS 10 IV. STABILITY OF THE PREDICTOR, CORRECTOR AND ALGORITHM 25 V. DISCUSSION OF STARTING TECHNIQUES AND CHANGE OF STEP-SIZE 42 VI. NUMERICAL EXAMPLES 46 VII. SUMMARY 50 VIII. BIBLIOGRAPHY 51 IX. ACKNOWLEDGMENT 52 X. APPENDIX 53 1 1. INTRODUCTION !n this paper we are considering an algorithm for the solution of quasi-linear hyperbolic partial differential equations by the method of characteristics. The method of characteristics is playing an increasingly important role in the engineering sciences and particularily in propagation type problems. The propagation problems arising in the theory of gas dynamics, the theory of long water waves and the theory of plasticity, to name a few, are well suited to the method of characteristics. The restriction to quasi-linear hyperbolic partial differential equations is understandable in view of the dependence of the method of characteristics on two sets of real characteristics. In this work, the method of characteristics is considered only for problems in two independent variables. Although this limitation is im­ plicit in nearly all practical applications of the method, there do exist certain procedures that utilize the special features of the characteristics for problems in more than two independent variables. The method of characteristics reduces the partial differential equation to a family of initial value problems. And although a substan­ tial knowledge of ordinary differential equations is available the need for discrete variable methods is well known. Thus the need for a high accuracy numerical method that can be easily applied to a wide range of practical problems seems to be present. Chapter two deals with the conversion of the quasi-linear,- second order, hyperbolic partial differential equation into characteristic normal form. The equivalence of the two problems is remarked upon and the 2 existence of a solution to the characteristic normal form system along with the differentiability of such solutions is proved. Chapter three investigates the truncation error involved in the predictor and the corrector. We also affirm the existence and uniqueness of solutions of these two systems used to approximate the true solution of the characteristic normal form system. In chapter four we discuss the stability or lack of stability of the predictor and corrector as well as the stability of the algorithm. The order and the degree of the algorithm is also defined here. A starting procedure is recommended in chapter five in addition to a procedure for altering the step-size of the characteristic mesh. Chapter six illustrates the algorithm with a numerical example. II. DISCUSSION OF THE ALGORITHM The second order, quasi-linear, hyperbolic partial differential equation as it arises in most practical situations has the form (2.1) au** + + cUyy + e = 0, 2 2 2 where ^ . "yy = H and the coefficients a, b, c dX dy and e, due to the quasi-linearity,are functions of x, y, u, p and q; (p = u^, q = Uy). We mention that due to the hyperbolic nature of the equation we are restricted to a region R of the xy-plane where (2.2) - ac > 0. We will however insist on a slightly more restrictive condition, namely that we stay within a region R' where the discriminant is bounded away from zero: (2.3) b^ - ac > d > 0, for d some constant. We also assume that along an initial curve, (2.4) y = f(x), a < X < b, u, p and q are known and are given by 4 (2.5) u = u^fx), p = p^fx), q = q^Cx) Furthermore we assume the initial curve is not a characteristic. Though the partial differential equation arising in practice is generally of the form (2.1) we find this particular form inappropriate for a formal discussion of convergence, stability and other facets connected with its solution. For this reason we will convert equation (2.1) to its so-called "characteristic normal form," 5 i = 1(1)3 k=l (2.6a) 5 ik 21 Î = 4(1)5, I ' sn k=l wi th 12 3 4 5 s «= X, s = y, s = u, s = p, s = q, and (2.6) 1 0 0 0 e_ 0 0 1 X. a (2.6b) (a'k) = -p •q 1 0 0 -X. 1 0 0 0 e^ 0 0 1 X. a b + /b^ - ac where X = b - /b^ - ac 5 it is well known ([3], [13]) that a great number of hyperbolic problems in two independent variables - including equation (2.1) as well as the general second order equation (2.7) (i(X, y. f, fy. «xy, fyy) ^ 0 " may be reduced to a characteristic normal form. We will be assuming that the data from the initial curve (2.4)/(2.5) is prescribed on the line (2.8) Ç + n = 1 in the Çn~plarie. i.e. x = x^(g, 1-Ç), y = y^ (Ç, 1-Ç), u = u^(Ç, 1-Ç), P = P](Ç> , q = A](Ç, 1"Ç)I 0 ^ Ç ^ 1. We will refer to the initial curve as (2.9) . c = {(Ç, n)I g + n = 1» 0£Ç£i, 0£n£i Since new characteristic coordinates can be introduced by replacing Ç with any function of Ç and by replacing n with any function of ri, and since the initial curve is not itself a characteristic there is no actual loss of generality in taking it to be of the special form (2.8). it has been shown [7] that equation (2.1) and the characteristic normal form (2.6) are equivalent. In the characteristic normal form (2.6) the characteristics emanating from the initial curve (2.8) are lines 6 (2.10) Ç=Cj, n = 1 - Cj, 0<_C^ £1. For any high-order finite difference algorithm we must be certain that the solutions of (2.6)/(2.8) are sufficiently differentlable. To establish the needed differentiability we adapt a more general theorem by A. Douglas [5]. Theorem 1.1: (A. Douglas). Assume a) a'^(S) e c^ for S = (s\ s^, s^, s^, s^)eU, b) [det(a"^(S) )| ^d > 0 for S e U, c) s*^ e c^, k = 1(1)5, and "S e U on the initial curve c, where U is a closed region of the 5-dImensional space and let D be defined as the triangular region enclosed by the initial curve c and the 1ines Ç = 1, n = '. * * Then in a certain region D CD, with D containing the initial curve c there exists a unique set of solutions s^, k = 1(1)5 of (2.6)/(2.8) which are four-times differentiable with respect to Ç and r). in view of later applications we will assume D to have the following properties (which are made possible by proceding to a subregion of the it original D ): a) D IS closed b) Given some £ > 0 (2.11) "s' = (s^(Ç, n) + . s^(Ç, n) + e^) £ U if (Ç, n) G D ' and £ e , k = 1(1)5. In the following we will assume that an e >0 has been chosen and that 0* is the region just described. Within this region we are investi­ gating the numerical solution of the problem. The predictor-corrector algorithm we now introduce to solve the system (2.6) with initial conditions described on c is based upon approximating the ten partial derivatives in (2.6) by forward and backward differences. We impose a square mesh of side h on the triangular region D. The calcu­ lated value at a point (Ç^ ) of the characteristic grid depends upon six back points, three along each of the characteristics 5 = and n = Hj. The partial derivatives and — will act as ordinary derivatives OS when applied along a line Ç = c^ or ri = C2. Therefore we will use the following well known ([10],[11]) forward and backward difference operators, truncated after third differences. (2.12a) f'(x) = 1 ^ f(*) h i=l (2.12) (2.12b) f ' (x) = 1 } j. V' f(x) . 1 =1 We also note the use of when the forward and backward operators are applied to the first variable in a function, F(Ç, ri) and when they are applied to the second variable in the argument; i.e.
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