Showing papers on "Hyperbolic partial differential equation published in 1978"

Numerical solution of partial differential equations : finite difference methods

Abstract: Substantially revised, this authoritative study covers the standard finite difference methods of parabolic, hyperbolic, and elliptic equations, and includes the concomitant theoretical work on consistency, stability, and convergence. The new edition includes revised and greatly expanded sections on stability based on the Lax-Richtmeyer definition, the application of Pade approximants to systems of ordinary differential equations for parabolic and hyperbolic equations, and a considerably improved presentation of iterative methods. A fast-paced introduction to numerical methods, this will be a useful volume for students of mathematics and engineering, and for postgraduates and professionals who need a clear, concise grounding in this discipline.

The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types

Abstract: Collapse in finite time is established for part of the solutions of certain classes of quasilinear equations of parabolic and hyperbolic types, the linear part of which has general form. Certain hyperbolic equations having L-M pairs belong to these classes.

Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation

Abstract: The nonlinear Boltzmann equation for a rarefied gas is investigated in the fluid dynamical limit to the level of compressible Euler equation locally in time, as the mean free path e tends to zero. The nonlinear hyperbolic conservation laws obtained as the limit are also the first approximation of the Chapman-Enskog expansion.

Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations

Abstract: Publisher Summary This chapter reviews some existence theorems for critical points of a real-valued function on a real Banach space and to apply these results to elliptic and hyperbolic partial differential equations The abstract results on critical points are obtained using minimax arguments Applications to elliptic equations are thereafter provided for the same A new proof is given for a recent result of Ahmad et al , as well as some variants of their result The work on abstract results on critical points is applied to hyperbolic problems

The Fourier method for nonsmooth initial data

Abstract: Application of the Fourier method to very general linear hyperbolic Cauchy problems having nonsmooth initial data is considered, both theoretically and computationally. In the absence of smoothing, the Fourier method will, in general, be globally inaccurate, and perhaps unstable. Two main results are proven: the first shows that appropriate smoothing techniques applied to the equation gives stability; and the second states that this smoothing combined with a certain smoothing of the initial data leads to infinite order accuracy away from the set of discontinuities of the exact solution modulo a very small easily characterized exceptional set. A particular implementation of the smoothing method is discussed; and the results of its application to several test problems are presented, and compared with solutions obtained without smoothing. Introduction. In recent years the Fourier method for the numerical approximation of solutions to hyperbolic initial value problems has been used quite successfully. In fact, if the initial function is C°° and the coefficients of the equation are constant the method converges arbitrarily fast, i.e. is limited in practice only by the method of time discretization which is chosen. This is the reason that the Fourier method is caled "infinite order" accurate. However, the situation is drastically different when the initial function is not smooth. We take as a model the one space dimension scalar problem ut = ux to be solved for 2ir periodic u on the interval n < x < n with initial values

Accuracy and Resolution in the Computation of Solutions of Linear and Nonlinear Equations

Abstract: Publisher Summary This chapter describes the accuracy and resolution in the computation of solutions of linear and nonlinear equations. In many problems, one is presented with piecewise initial data whose discontinuities occur along surfaces. According to the theory of hyperbolic equations, solutions with such initial data are themselves piecewise with their discontinuities occurring across characteristic surface. At points away from the discontinuities, the truncation error is small. In these regions, it is reasonable to use difference approximations of high order accuracy, except for the danger that the large truncation error at the discontinuities propagates into the smooth region. According to the theory of hyperbolic conservation laws, solutions of systems of the form are in general discontinuous. The discontinuities, called shocks, need not be present in the initial values but arise spontaneously and their speed of propagations is governed by the Rankine–Hugoniot jump relation. It is more difficult to construct accurate approximations of discontinuous solutions of nonlinear equations than of linear equations.

Nonlinear Partial Differential Equations and Applications

Abstract: Abstract: Fluid particles like drops or bubbles play a prominent role in numerous applications like multiphase chemical reactors, fuel engines, atomization, drying of liquid sprays, heat exchange and ink-jet printing to mention just a few. One particular development towards process intensifications relies on micro-systems, which further enhances the role of intermaterial interfaces and requires accurate models and a profound understanding of these. This talk surveys the continuum mechanical modeling of two-phase flows with increasing level of physico-chemical interface properties, starting from a simple dividing interface to the case when the interface is a phase for itself with surface viscosity and variable surface tension the so-called BoussinesqScriven surface fluid. For the different levels of interfacial properties, the corresponding mathematical models together with main analytical results are outlined. Recent results on numerical approaches are included for some of the models, where main emphasis is put on the Volume-ofFluid (VOF) method.

Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients

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A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations

Abstract: Superconvergence phenomena are demonstrated for Galerkin approxima- tions of solutions of second order parabolic and hyperbolic problems in a single space variable. An asymptotic expansion of the Galerkin solution is used to derive these results and, in addition, to show optimal order error estimates in Sobolev spaces of negative index in multiple dimensions. 1. Introduction. We shall be concerned primarily with the analysis of supercon- vergence phenomena associated with the numerical solution of second order, linear parabolic and hyperbolic equations by Galerkin methods based on piecewise-polyno- mial spaces. Our principal tool will be an asymptotic expansion to high order of the Galerkin solution; this expansion will be obtained by using a sequence of elliptic pro- jections and will be called a quasi-projection. In Sections 4 and 5 we develop the quasi-projection for parabolic Galerkin procedures for problems in one or several space variables for both Neumann and Dirichlet boundary conditions and derive optimal order negative norm estimates for the error in the Galerkin solution. In Section 6 we apply the quasi-projection to de- rive superconvergence results in the case of a single space variable when the Galerkin space consists of piecewise-polynomial functions of degree r. It is well known (4), (6), (7), (9), (10) that, if h is the knot spacing parameter associated with the not necessarily uniform grid, the Galerkin solution for standard parabolic problems con- verges with an error that is at best globally of order 0(hr+ ), as measured in L2 or L°°. Consider a knot at which the smoothness constraint of the Galerkin space re- duces to continuity. We show that the Galerkin solution produces an 0(h2r)-ap- proximation at such a knot. Also, we show that a very simply evaluated weighted quadrature of the Galerkin solution gives an cXft2^-approximation of the flux at the knot; the direct evaluation of the derivative of the Galerkin solution leads to an 0(/1r)-approximation. We summarize briefly in Section 7 results presented in detail elsewhere (3) showing that the superconvergence results above are preserved and that supercon- vergence occurs in the time increment when the Galerkin procedure is discretized in time by a collocation method. In Section 8 we treat continuous-time Galerkin methods for hyperbolic prob- lems and obtain analogous results. Throughout this paper we rely heavily on some earlier results of two of the

On the convergence of solutions of hyperbolic equations

Abstract: (1978). On the convergence of solutions of hyperbolic equations. Communications in Partial Differential Equations: Vol. 3, No. 1, pp. 77-103.

Double Walsh series solution of first-order partial differential equations

Abstract: A double Walsh series is introduced to represent approximately functions of two independent variables, and is then applied to analyse single as well as simultaneous first-order partial differential equations. The solutions for the coefficient matrices can be obtained directly from Kronecker product formulae, which are suitable for computer computation. An example for a single first-order partial differential equation is solved by a double Walsh series approximation with satisfactory results.

Exact, multiple soliton solutions of the double sine Gordon equation

Abstract: Exact, particular solutions of the double sine Gordon equation in $n$ dimensional space are constructed. Under certain restrictions these solutions are $N$ solitons, where $N\leq 2q-1$ and $q$ is the dimensionality of space-time. The method of solution, known as the base equation technique, relates solutions of nonlinear partial differential equations to solutions of linear partial differential equations. This method is reviewed and its applicability to the double sine Gordon equation shown explicitly. The $N$ soliton solutions have the remarkable property that they collapse to a single soliton when the wave vectors are parallel.

A new method to predict swelling using a hyperbolic equation

Abstract: Light buildings founded on expansive soils may be subjected to undesirable cracking resulting from differential swelling and/or shrinking of the soil. The prediction of probable heave of these soils under given surcharge loads is therefore becoming important in day-to-day soil engineering practice. Several detailed investigations have been made in the past to characterise swelling-swelling pressure-time relationships for expansive clays. These relationships and other methods to predict swelling have had only a limited success since problems involving swelling behaviour are complex and can only be solved by an approach based on the observed properties of these soils. An attempt is made in this study to predict the maximum swelling from the observed behaviour over a short interval of time. The approach is based on a hyperbolic equation. The test data obtained from several tests have been analysed to examine the validity of the hyperbolic equation. Published data have also been analysed to verify the validity of the equation and the analyses show a good agreement between predicted and observed maximum swelling. /Author/TRRL/

Boundary Conditions for Limited-Area Forecasts

Abstract: The smoothness of solutions of the barotropic and baroclinic filtered and primitive equations in limited areas, subject to inflow boundary conditions, is in doubt due to the necessary occurrence of points on the boundaries where the flow is tangential. The problem is described in detail for the barotropic filtered equations; it is the classical situation of Cauchy data being specified for a hyperbolic equation on a boundary curve which is locally characteristic. A necessary set of conditions is developed for the data which must be satisfied if a smooth solution is to be ensured. The set is far too complex to allow their satisfaction in practice. Only the objectionable practice of introducing very high viscosity near the boundaries provides a simple way out of the problem. A more reasonable and not too complicated alternative is suggested.

The Hodie Method and its Performance for Solving Elliptic Partial Differential Equations

Abstract: Publisher Summary This chapter describes the Hodie method and its performance for solving elliptic partial differential equations. It discusses a new flexible, high-accuracy finite difference approximation to the elliptic partial differential equation. A rectangular mesh is put and at each mesh point an estimate is obtained as the solution of a finite difference equation. For simplicity of exposition, the chapter presents an assumption that the mesh is uniform with mesh spacing and this assumption is not essential to the method though it improves its efficiency in some cases. It is found that after the auxiliary points are chosen, the coefficients of the difference equation are determined to make the approximation exact on a given linear space of functions. The chapter presents a general discussion of the method's computational properties and potential applicability along with a comparative performance evaluation using the ELLPACK system. The usual difference equation for a second order problem can be derived by making the scheme exact on quadratic polynomials and it is automatically exact on cubic polynomials. It is found that the order of the discretization error is the same as the order of the truncation error.

A generalised cole-hopf transformation for nonlinear parabolic and hyperbolic equations

Abstract: The Cole-Hopf transformation has been generalized to generate a large class of nonlinear parabolic and hyperbolic equations which are exactly linearizable. These include model equations of exchange processes and turbulence. The methods to solve the corresponding linear equations have also been indicated.

A comparison of the Method of Lines to finite difference techniques in solving time-dependent partial differential equations. [with applications to Burger equation and stream function-vorticity problem]

Abstract: Abstract Steady state solutions to two time dependent partial differential systems have been obtained by the Method of Lines (MOL) and compared to those obtained by efficient standard finite difference methods: (i) Burger's equation over a finite space domain by a forward time—central space explicit method, and (ii) the stream function—vorticity form of viscous incompressible fluid flow in a square cavity by an alternating direction implicit (ADI) method. The standard techniques were far more computationally efficient when applicable. In the second example, converged solutions at very high Reynolds numbers were obtained by MOL, whereas solution by ADI was either unattainable or impractical. With regard to “set up” time, solution by MOL is an attractive alternative to techniques with complicated algorithms, as much of the programming difficulty is eliminated.

The Cauchy problem for hyperbolic operators with variable multiple characteristics

Abstract: Let P(t, x, D„ Dx) be a hyperbolic differential operator with the principal symbol pm(t, x, t, Q. We assume that Pm is denoted by IÇ_ i(t W&flYi* \) **d /»|/-o = % U ~ 1» • • • > m ~ 1) is well posed. When m¡ — 1 (J ■ 1.j), our result coincides those of Ohya and Petkov. 1. Statement of the result. We shall consider the following differential operator: P(t,x,D„Dx)= S aa(t,x)Dr*DÏ, \a\/ = A"0^",' ■ • • A? and Z), = z'8/3i, Djj. = id/dxj. We assume that m2> • • • > ms, N = 2*_i wy and all functions Xj(t, x, £) (J = 1,. .., m N + s) are real and positively homogeneous of degree 1 with respect to £ and belong to C, (r, x, |) E [0, T] X Ä" X S"-\ where C is a positive constant. Throughout this note, the symbols of pseudo-differential operators are elements of <$([0, T] X R" X 5") or $([0, T] X 7T X S"-1) if (t, £) E 5" or Received by the editors May 13, 1977 and, in revised form, August 8, 1977. AMS (MOS) subject classifications (1970). Primary 35L30; Secondary 35L45. © American Mathematical Society 1978 109 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 110 KAZUHIRO YAMAMOTO £ G S"1-1. Moreover we use the following notation. For symbols a(t, x, |), b(t, x, £) and A(t, x, t, £), A\T=a = 0 mod b means that there exists a symbol c(t, x, £) such that A(t, x, a(t, x, £), Í) = c(t, x, £)b(t, x, £). For the lower order terms of P we assume the following: (A.3) For any k (k = 1,.. . , s) we can denote P(t, x, D„ Dx) by the following form: mk P=IlQkJ(t,x,Dl,Dx)(Ak(t,x,D„Dx)). (1.1) 1=0 Here Ak is the pseudo-differential operator defined by the symbol t A^f, x, £) and Qkl (1 = 0,..., mk) is a pseudo-differential operator of order m mk, whose principal symbol qkj(t, x, t, £) has the following property: ?M^e0 modXk Xm_N+k. (1.2) Clearly if Xk = Am_Ar+fc, then the above condition (A.3) is that of E. E. Levi. In the final part of this note we denote (A.3) by the condition with respect to pk, when mk = 1 or 2. For a nonnegative integer k and s E R the function space C*([0, T]; HS(R")) consists of functions such that D{u(t) (j = 0, ..., k) exists as an element of Hs_j(R") and is continuous on the topology of Hs_j(R"). We use the following norm: IIKOIIft* = 2 l|£/«(0ll?-y> 7 = 0 where || • \\s_j is the usual norm of Hs_j(R"). Now we can state our theorem. Theorem. Let P(t, x, D„ Dx) be a differential operator of order m. If P satisfies the assumptions (A.l), (A.2) and (A.3), then the Cauchy problem Pu = fin [0, T]X R", D/«|/=,0 = gj(J = 0, . . .,m I) is well posed, i.e., for f 6 C*—+"'+,aft T); Hs-m+mi + x(Rn))andgj E Hs_j+m(R») there exists a unique solution u(t, x) G C*([0, T]; HS(R")) such that ( m-\ IIKOIIU < c 2 llg,IL-,+mi + lll/(0)|IL-m+mi,A_m+mi I 7=0 • illl/tollL-m+m. + U-m+m. + l^l. wAere k > m m, 1, |||/(0)|||J_m+m,>_1 = 0 am/. G [0, T]. This Theorem is the same as those of [3] and [4] when m, = 1 (J = 1,. .., s). Under a different situation, in [1] they consider the Cauchy problem of a triple case. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE CAUCHY PROBLEM FOR HYPERBOLIC OPERATORS 111 2. Reform of the condition (A3). In this section, we state an equivalent condition of (A.3). Taking care of multiplicities of the roots X¡ (j = 1,. . ., s), we can denote pm(t, x, t, £) by (t \„_N+S) ■ ■ -(r-Xs+xWr ■■&,>, where d>„(f, x,t,E)(v = 1, ..., u) is a polynomial of degree s with respect to t and equal to II^.i(t Xj). Here ms¡ + x = • • • — m, (v — 1, . . . , u). Remark that sx < s2 < ■ ■ ■ < s^ = s, mx = 2¡;=1/z„ and denote N„ by svnp. We introduce a product pseudo-differential operator $„(/, x, Dt, Dx) = (Aj • • • A,)(/, x, Dt, Dx). Then we denote Ay(t, x, D„ Dx) of order j (J = 0,..., m) by A, 1, A, A„ ..., A,,..., àN Of• • *?',..., A„+, = A1+t • • • AJ+1A^, . . . , Am = Am_„+i • • • AJ+1AN, where A, = As • • • A,W_j •••$;' if y = (Nx + • • • + N„_x) + os„ + 8 (o = 0,.. ., n, — 1,8 = 0,..., sy_x). Then we have the following: Proposition 2.1. Let P(t, x, D„ Dx) be a differential operator which satisfies the conditions (A.1) and (A.2). Then the condition (A.3) is equivalent to the following statement. We can denote P by 771, P (t, x, D„ Dx ) = 2 Q, ÍU x, D„ Dx )A/((> (2.1) ( = 0 where if i = «M + • • • + w„+1 + o (1 < a < n„), then I(i) = Nx + • • • + iV„ — s„o and Q¡ (i = 0, . . ., m,) is a pseudo-differential operator of order Mi = m — i — I (i) and differential operator of t. Moreover the principal symbol q¡(t, x, t, £) of Q¡ satisfies the following condition: (7,1^=0 mod A* -K-s+k ifk*>¿>,)A/0)+,, (2-3) where R¡j is a pseudo-differential operator of order M¡ — j whose principal symbol is r¡j(t, x, £), end rtJ have the following property: k-\ 2 rtJAj ■ • • A,,,^ 0 mod A* X„,_N+k, k < s„ (2.4) y-o if i = zzM + • • • + rt„+1 + o (1 < a < zz„). 3. Reduction to a first order system and the proof of the Theorem. Since the proof of the Theorem is inferred on the analogy of a simple case, we assume that j = 2, ttz, = 2 and m2 = 1. Thus by (2.3) our considered operator License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 112 KAZUHIRO YAMAMOTO P(t, x, D„ Dx) in (1.1) is denoted by 2 m — i p(t,x,D„Dx) = Am + '2 2*r'A> í-1 j=0 where Rf"~' is of order m — i — j. The condition (2.4) says that R¿-l(t,x,Dx) = 0, (3.1) ri""1 (/, x, ¿) = r0—2(/, x, |) = 0 mod X, \„_2, (3.2) (ri"-1 + r2—' (X2 Xx))(t, x,^) = 0 mod X2 X„,_x. (3.3) We denote a pseudo-differential operator with the symbol |£| by the term A and define a column vector U ='(Am-3V> Am"3A,«, Am-4A2w, Am-4A3M,...,AAm_2M,Am_lM) (3.4) and F '(0,..., 0,/). Then by (3.1) the equation Pu = / becomes the following first order system: MU = (D, ¿(/, x, DX))U + B(t, x, DX)U = 77, where 5 is of order 0 and A is a first order pseudo-differential operator with the symbol

An Experimental Design for the Computational Evaluation of Elliptic Partial Differential Equations Solvers

Abstract: 1. Introduction. In this report we present a preliminary a priori experi

Numerical analysis of pressure transients in bubbly two-phase mixtures by explicit-implicit methods

Abstract: The one-dimensional equations for transient two-phase flow are a system of nonlinear hyperbolic partial differential equations, expressible, under certain assumptions, in conservation form. Inasmuch as the use of the method of characteristics becomes complicated if shock waves are present, it is easier to follow a gas-dynamics approach and employ one of the available procedures for solving one-dimensional systems of conservation equations. A recently introduced technique, due to McGuire and Morris [1, see also 12] and known as an Explicit-Implicit method, is used here for a simple boundary-value problem of wave propagation in bubbly two-phase mixtures, and is found to be simple and versatile. A comparison of this method with the well-known Lax-Wendroff (two-step) scheme demonstrates that shock fronts are simulated better, oscillations behind the shocks are smoothable by parameter adjustment, and computation time is reduced when the Explicit-Implicit method is employed.