Engineering Transactions, 17, 3, pp. 485-529, 1969

### O metodach przybliżonego rozwiazywania problemów falowych w ośrodkach niesprężystych

A. Pielorz
Instytutu Podstawowych Problemów Techniki
Poland

In the paper we have confined ourselves to those wave problems which can be brought down to the problem of solving the boundary-value problem for hyperbolic sets of partial differential equations of the first order. In general such equations are not integrable in closed form. The methods
of solution used are approximate methods, most often the methods of finite differences. In the case of utilizing the method of finite differences instead of a set of differential equations we solve a set of difference equations. The necessary condition for the convergence of an approximate solution to the exact solution is the stability of the set of difference equations. Chapter 2 contains the best
known definitions of stability for a linear set of difference equations with constant and variable coefficients.
The major purpose of this paper is to compare five methods of approximate solution of sets of partial differential equations of the first order of hyperbolic type with two independent variables. These are: the iteration method of Courant, method of direct integration, method of finite differences along the characteristics, the Keller-Thomée method and iteration methods which can be applied
only in the case of such sets of equations which can be reduced to a partial differential equation of the second order. These methods have been described in Chapters 3-7. Iteration methods and Keller-Thomée method ensure the convergence of the approximate solution to the exact solution of the differential equations. Courant's method, the method of direct integration and the method of finite differences along characteristics have been compared on the example of a cylindrical shear wave in an elastic/viscoplastic medium. In the case of Courant's method 3 iterations were calculated. The numerical results are characterized by good conformity.
The problem of stability of the methods considered has been discussed. The iteration methods are analytical methods and for them the problem does not exist. For the Keller-Thomée method the absolute stability has been demonstrated in [28]. For the remaining two methods the stability problem is open. The stability of these two methods has been analyzed in the case of solving a set of two semi-linear Eqs. (8.1), describing the problem of propagation of a cylindrical shear wave in an elastic/viscoplastic medium. Taken there as a basis were the papers [23] and [55]. It has been shown that in the case of the method of finite differences along the characteristics the definition of stability described in Chapter 2 cannot be utilized.
Chapter 10 contains some considerations concerning the continuity of derivatives of the solution for the cylindrical shear wave in an clastic/viscoplastic on a load curve,

Full Text: PDF

#### References

A. K. AZIZ, J.B. DIAZ, On a mixed boundary-value problem for linear hyperbolic partial differential equations in two independent variables, Arch. Rat. Anal., 10 (1962), 1-28.

J. BEJDA, Propagation of two-dimensional stress waves in an elastic/viscoplastic materials, Procecdings of 12th International Congress of Applied Mechanics held at Stanford in 1968.

И. С. Березин, H. П. Жидков, Методы вычислений, T, 2, Москва 1962.

M. BURNAT, Theory of simple waves for nonlinear systems of partial differential equations of the first order and applications to gas dynamics, Arch. Mech. Stos., 4, 18 (1966). 5. M. BURNAT, A. KIELBASINSKI, Computation of a three dimensional flow, Fluid Dynamics Transations, 3 (1967).

M. BURNAT, A. KIELBASINSKI, A. WAKULICZ, The method of characteristics for a multi-dimensional gas flow, Arch. Mech. Stos., 3, 16 (1964).

D.S. BUTLER, The numerical solution of hyperbolic systems of partial differential equations in three independent variables, Proc. Roy. Soc., London 1960.

C.S. CHU, J. B. DIAZ, Remarks on a mixed boundary-value problem for linear hyperbolic partial differential equations in two independent variables, Arch. Rat. Mech, Anal., 3, 16 (1964), 187-195.

R. J. CLIFTON, Elastic-plastic boundaries in combined longitudinal and torsional plastic wave propagation, report. 10. E. CONWAY, J. SMOLLER, Global solutions of the Cauchy problem for quasi-linear first order equations in several space variables, Comm. Purc Appl. Math., 1, 19 (1966), 95-105.

R. COURANT, Cauchy's problem for hyperbolic system of first order partial differential equations in two independent variables, Comm. Pure Appl. Math., 3, 14 (1961), 257-265.

R. COURANT, D. HILBERT, Methods of mathematical physics. T. 2, Partial differential equations by R. Courant, Interscience Publishers, Now York-London 1962.

R. COURANT, W. ISAACSON, M. REES, On the solution of non-linear hyperbolic differential equations by finite differences, Comm. Pure Appl. Math., 5 (1952), 243-255.

J. T. FONG, Elastic-plastic wave in a half-space of a linearly work-hardening material for coupled shear loadings, report 1966.

J. T. FONG, On the generation of strong discontinuities in a moving boundary-value problem in dynamic plasticity, report.

G. E. FORSYTHE, W.R. WASOW, Finite-difference methods for partial differential equations, New York-London 1960,

P. Fox, The solution of hyperbolic partial differential equations by difference methods, Mathematical methods for digital computers, New York-London 1960 (Ralston ed.).

K.O. FRIEDRICHS, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., 7 (1954), 345-392.

J. GLIMM, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 4, 18 (1965), 697-716.

С. К, Годунов, В, С. Рябенький, Введение в теорию разностных схем, Москва 1962.

A.R. GOURLAY, A.R. MITCHELL, Alternating direction methods for hyperbolic systems,Num. Math., 2, 8 (1966),

A. R. GOURLAY, A. R. MITCHELL, A stable implicit difference method for hyperbolic systems in two space variables, Num. Math., 4, 8 (1966).

S. G. HAHN, Stability criteria for difference schemes, Comm. Pure Appl. Math., 11 (1958), 243 -255.

E. ISAACSON, Fluid dynamical calculations. Numerical solutions of partial differential equations, Maryland 1965, 35-48.

F. JOHN, On integration of parabolic equations by difference methods, Comm. Pure Appl. Math.,

(1962), 155-211.

Л. В. Канторович, Г. П. Акилов, Функциональный анализ в нормированных простран-ствах, Москва 1959.

H. B. KELLER, On the solution of semi-linear hyperbolic systems by unconditionally stable difference methods, Comm. Pure Appl. Math., 14 (1961), 447-456.

H.B. KELLER, V. THOMÉE, Unconditionally stable difference methods for mixed problems for quasi-linear hyperbolic systems in two dimensions, Comm. Pure Appl. Math., 15 (1962), 63-73.

Z. KOWALSKI, A difference method for the non-linear partial differential equation of the first order, Annal. Pol. Math., 3, 8 (1966).

, Z. KOWALSKI, A difference method for certain hyperbolic systems of non-linear partial differential equations of the first order, Annal. Polon. Math., 19 (1967), 313-322.

H.O. KREISS, Über die stäbilitätsdefinition für differenzengleichungen die partielle differential-gleichungen approximieren, BIT, 2 (1962), 153-181.

F. O. KREISS, On difference approximations of the dissipative type for hyperbolic differential equations, Comm. Pure Appl. Math., 17 (1964), 335-353.

H. O. KREISS, Difference approximations for hyperbolic differential equations, Numerical solution of partial differential equations, Maryland 1965.

, P. D. LAX, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math., 7 (1954), 159-193.

P.D. LAX, Hyperbolic systems of conservation laws, Comm. Pure Appl. Math., 4, 10 (1957) 537-566.

P. D. LAX, On the stability of difference approximations to solutions of hyperbolic equations with variable coefficients, Comm. Pure Appl. Math., 14 (1961), 497-520.

P.D. LAX, Numerical solution of partial differential equations, Ann. Math. Monthly, 2, 72 (1965), 74-84.

P. D. LAX, R. D. RITCHMYER, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math., 9 (1956), 267-293.

P. D. LAX, B. WENDROFF, Systems of conservation laws, Comm. Pure Appl. Math., 2, 13 (1960), 217-238.

P.D. LAX, B. WENDROFF, On the stability of difference schemes, Comm. Pure Appl. Math., 15 (1962), 363-371.

P. D. LAX, B. WENDROFF, Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math., 17 (1964), 381-398.

M. LISTER, The numerical solution of hyperbolic partial differential equations by the method of characteristics, Mathematical methods for digital computers, New York-London 1960.

О. А. Ладыженская, Решение задачи Коши для гиперболических систем методом ко-нечных разностей, УЗ ЛГУ, серия мат, наук, 144, 23 (1952), 192-246.

G. MAJCHER, Sur un probleme mixte pour l'équation du type hyperbolique, Annal. Polon. Math., 5 (1958), 121 -133.

B. PARLETT, Accuracy and dissipation in difference schemes, Comm. Pure Appl. Math., 19 (1966), 111-123.

P. PERZYNA, Propagation of shock waves in non-homogeneous elastic/visco-plastic bodies, Arch. Mech. Stos., 6, 13 (1961), 851-867.

P. PERZYNA, On A non-linear boundary-value problem for 0 linear hyperbolic partial differential equation, Bull. Acad. Polon. Sci., Série Sci. Techn., 12 (1964), 589-594.

P. PERZYNA, The application of the iteration method to the solution of the problems of propagation of stress waves in an inelastic medium, Arch. Mech. Stos., 1, 17 (1965), 87-107.

P. FERZYNA, Teoria lepkoplastyczności, Warszawa 1966.

P. PERZYNA, J. BEJDA, The propagation of stress waves in a rate sensitive and work-hardening plastic medium, Arch. Mech. Stos., 6, 16 (1964), 1215-1244.

P. PERZYNA, A. PIELORZ, Discussion of methods of approximate solution of wave problem in an inelastic medium, Arch. Mech. Stos., 1, 19 (1967), 115-127.

A. PIELORZ, The comparison of methods for solving wave problems in inelastic media, Arch. Mech. Stos., 2, 20 (1968), 225-241.

G. PROUSE, Sulla risoluzione del problema misto per le equazioni iperboliche non lineari mediante le differenze finite, Ann. di Mat., 46 (1958), 313-341.

R. D. RITCHMYER, Difference methods for initial-value problems, New York 1957.

В. С. Рябенький, А. Ф. Филиппов, Об устойчивости разностных уравнений, Москва 1956.

W. W. RUSANOV, Difference methods of constant direction, Arch. Mech. Stos., 6, 20 (1968).

H. SAUERMAN, M. SUSSMAN, Numerical stability of three-dimensional method of characteristics, AIAA Journal, 2, 2 (1964), 387-390.

W. G. STRANG, Difference methods for mixed boundary-value problems, Duke Math. J., 27 (1960), 221-23I.

Z. SZMYDT, Sur un nouveau Lype de problèmes pour un système d'équation différentielles hyper-boliques du second ordre à deux variables indépendantes, Bull. Acad. Polon., CI. IIT, 4 (1956), 67-72.

Z. SZMYDT, Sur une généralisation des problemes classiques concernant un système d'équations différentielles hyperboliques du second ordre à deux variables indépendantes, Bull. Acad. Polon., CI. III, 4 (1956), 579-584.

Z. SZMYDT, Sur un problème concernant un système d'équations différentielles hyperboliques d'ordre arbitraire à deux variables indépendantes., Bull. Acad. Polon., Cl. III, 5 (1957), 577-582.

Z. SZMYDT, Sur le problème de Goursat concernant les équations différentielles hyperboliques du second ordre, Bull. Acad. Polon., CI. TII, 5 (1957), 571-575.

Z. SZMYDT, Sur l'éxistence de solutions de certains problemes aux limitès relatifs à 74F1. Système d'équations différentielles hyperboliques, Bull. Acad. Polon., Série Sci. Math., Astr. et Phys., 6 (1958), 31-36.

V. THOMÉE, Difference methods for two-dimensional mixed problems for hyperbolic first order systems, Arch. Rat. Mech. Anal., 8 (1961), 68-88.

V. THOMÉE, A mixed boundary-value problem for hyperbolic first order systems with derivatives in the boundary conditions, Arch. Rat. Mech. Anal., 8 (1961), 435-443.

V. THOMÉE, A stable difference scheme for the mixed boundary problem for a hyperbolic first order systems in two dimensions, J. Soc. Indust. Appl. Math., 1962.

V. THOMÉF, A difference method for a mixed boundary problem for symmetric hyperbolic systems, Arch. Rat. Mech. Anal., 2, 13 (1963), 122-136.

В. Я. Урм, О необходимых и достаточных условиях устойчивости систем разностных уравнений, ДАН 1961, Т. 139, Na 1, 40-43.

B. WENDROFF, On centered difference equations for hyperbolic systems, J. Soc. Indust. Appl. Math., 8 (1960), 549 -555.

А. И. Жуков, Применение метода характеристик к численному решению одномерных задач газовой динамики, ТМИ им. В, А. Стеклова, 58 (1960).

В. В. Соколовский, Распространение цилиндрических волн сдвига в упруго-вязко-пласти-ческой среде, ДАН СССР, 60 (1948), 1325.

B. JASIŃSKI, Komunikat prywatny, 1967.