The exact region of stability for MacCormack scheme

Let the two dimensional scalar advection equation be given by $$\frac{{\partial u}}{{\partial t}} = \hat a\frac{{\partial u}}{{\partial x}} + \hat b\frac{{\partial u}}{{\partial y}}.$$ We prove that the stability region of the MacCormack scheme for this equation isexactly given by $$\left( {\hat a\frac{{\Delta _t }}{{\Delta _x }}} \right)^{2/3} + \left( {\hat b\frac{{\Delta _t }}{{\Delta _x }}} \right)^{2/3} \leqslant 1$$ where Δ _t , Δ _x and Δ _y are the grid distances. It is interesting to note that the stability region is identical to the one for Lax-Wendroff scheme proved by Turkel.     

Convergence,stability and truncation error estimation of a method for the numerical integration of the initial value problemY″=F(X,Y)

In this paper a detailed study of the convergence and stability of a numerical method for the differential problem $$\left\{ \begin{gathered} y'' = f(x,y) \hfill \\ y(x_0 ) = y_0 \hfill \\ y'(x_0 ) = y_0 ^\prime \hfill \\ \end{gathered} \right.$$ has carried out and its truncation error estimated. Some numerical experiments are described.     

Formule di quadratura «chiuse» di tipo gaussiano: Tabulzione dei nodi e dei coefficienti

The coefficientsA _hi and the nodesx _mi for «closed” Gaussian-type quadrature formulae $$\int\limits_{ - 1}^1 {f(x)dx = \sum\limits_{h = 0}^{2_8 } {\sum\limits_{i = 0}^{m + 1} {A_{hi} f^{(h)} (x_{mi} ) + R\left[ {f(x)} \right]} } } $$ withx _m0 =?1,x _{m, m+1} =1 andR[f(x)]=0 iff(x) is a polinomial of degree at most2m(s+1)+2(2s+1)?1, have been tabulated for the cases: $$\left\{ \begin{gathered} s = 1,2 \hfill \\ m = 2,3,4,5 \hfill \\ \end{gathered} \right.$$ .     

A note on the circular complex centered form

The centered form for real rational functions suggested by R. E. Moore [6] was extended to complex polynomials over circular complex domains in [7]. Here it is shown that the inclusion chain $$\begin{gathered} circular complex centered form evaluation \subseteq \hfill \\ Horner scheme \subseteq \hfill \\ power sum evaluation \hfill \\ \end{gathered} $$ is valid for all complex polynomials and all circular domains.     

Successive overrelaxation method with projection for finite element solutions applied to the Dirichlet problem of the nonlinear elliptic equation Δu-bu2

this study is a continuation of a previous paper [Computing 38 (1987), pp.117–132]. In this paper, we consider the successive overrelaxation method with projection for obtaining finite element solutions applied to the Dirichlet problem of the nonlinear elliptic equation $$\begin{gathered} \Delta u = bu^2 in\Omega , \hfill \\ u = g(x)on\Gamma . \hfill \\ \end{gathered} $$ . Some numerical examples are given to illustrate the effectiveness.     

Quantitative theorems on approximation by Bernstein-Stancu operators

In 1972 D. D. Stancu introduced a generalization \(L_{mp} ^{< \alpha \beta \gamma > }\) of the classical Bernstein operators given by the formula $$L_{mp}< \alpha \beta \gamma > (f,x) = \sum\limits_{k = 0}^{m + p} {\left( {\begin{array}{*{20}c} {m + p} \\ k \\ \end{array} } \right)} \frac{{x^{(k, - \alpha )} \cdot (1 - x)^{(m + p - k, - \alpha )} }}{{1^{(m + p, - \alpha )} }}f\left( {\frac{{k + \beta }}{{m + \gamma }}} \right)$$ . Special cases of these operators had been investigated before by quite a number of authors and have been under investigation since then. The aim of the present paper is to prove general results for all positiveL _mp ^<αβγ> 's as far as direct theorems involving different kinds of moduli of continuity are concerned. When applied to special cases considered previously, all our corollaries of the general theorems will be as good as or yield improvements of the known results. All estimates involving the second order modulus of continuity are new.     

A technical note on AC-Unification. The number of minimal unifiers of the equation αx1 + ⋯ + αxp ≐AC βy1 + ⋯ + βyq

We show in this note that the equation αx₁ + #x22EF; +αx_p?_ACβy₁ + α +βy_q where + is an AC operator and αx stands for x+...+x (α times), has exactly $$\left( { - 1} \right)^{p + q} \sum\limits_{i = 0}^p {\sum\limits_{j = 0}^q {\left( { - 1} \right)^{1 + 1} \left( {\begin{array}{*{20}c} p \\ i \\ \end{array} } \right)\left( {\begin{array}{*{20}c} q \\ j \\ \end{array} } \right)} 2^{\left( {\alpha + \begin{array}{*{20}c} {j - 1} \\ \alpha \\ \end{array} } \right)\left( {\beta + \begin{array}{*{20}c} {i - 1} \\ \beta \\ \end{array} } \right)} } $$ minimal unifiers if gcd(α, β)=1.     

Stability Criteria for Finite Difference Approximations to Parabolic Systems—An Update

In a recent series of papers, Goldberg [G1, G2] and Sun and Yuan [SY] studied the L ₂-stability of a well-known family of finite difference approximations for the initial-value problem associated with the multispace-dimensional parabolic system $$\frac{{\partial {\text{u(x,}}\;t{\text{)}}}}{{\partial t}} = \sum\limits_{1 \leqslant {\kern 1pt} p{\kern 1pt} \leqslant {\kern 1pt} q{\kern 1pt} \leqslant {\kern 1pt} {\kern 1pt} s} {A_{pq} \frac{{\partial ^2 {\text{u(x,}}\;t{\text{)}}}}{{\partial x_p \partial x_q }}} + \sum\limits_{1 \leqslant {\kern 1pt} p{\kern 1pt} \leqslant {\kern 1pt} s} {B_p \frac{{\partial {\text{u(x,}}\;t{\text{)}}}}{{\partial x_p }} + C{\text{u(x,}}\;t{\text{)}}} $$ where A _pq ,B _p and C are constant matrices, A _pq being Hermitian. In the present paper we discuss these earlier results and complete the underlying theory by answering four open questions.     

Sulle formule di quadratura di Tschebyscheff

For the Tschebyscheff quadrature formula: $$\int\limits_{ - 1}^1 {\left( {1 - x^2 } \right)^{\lambda - 1/2} f(x) dx} = K_n \sum\limits_{k = 1}^n {f(x_{n,k} )} + R_n (f), \lambda > 0$$ it is shown that the degre,N, of exactness is bounded by: $$N \leqslant C(\lambda )n^{1/(2\lambda + 1)} $$ whereC(λ) is a convenient function of λ. For λ=1 the complete solution of Tschebyscheff's problem is given.     

On the convergence rate of SOR: A worst case estimate

LetA be any real symmetric positive definiten×n matrix, and κ(A) its spectral condition number. It is shown that the optimal convergence rate $$\rho _{SOR}^* = \mathop {\min }\limits_{0< \omega< 2} \rho (M_{SOR,\omega } )$$ of the successive overrelaxation (SOR) method satisfies $$\rho _{SOR}^* \leqslant 1 - \frac{1}{{\alpha _n \kappa (A)}}, \alpha _n \approx \log n.$$ This worst case estimate is asymptotically sharp asn→∞. The corresponding examples are given by certain Toeplitz matrices.     

Branching Time Logics \mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha } with Operations Until and Since Based on Bundles of Integer Numbers, Logical Consecutions, Deciding Algorithms

This paper is intended as an attempt to describe logical consequence in branching time logics. We study temporal branching time logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ which use the standard operations Until and Next and dual operations Since and Previous (LTL, as standard, uses only Until and Next). Temporal logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ are generated by semantics based on Kripke/Hinttikka structures with linear frames of integer numbers $\mathcal {Z}$ with a single node (glued zeros). For $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ , the permissible branching of the node is limited by α (where 1≤α≤ω). We prove that any logic $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ is decidable w.r.t. admissible consecutions (inference rules), i.e. we find an algorithm recognizing consecutions admissible in $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ . As a consequence, it implies that $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ itself is decidable and solves the satisfiability problem.     

Esistenza,unicità e calcolo della soluzione di un sistema non lineare

LetX be a point of the realn-dimensional Euclidean space ? ⁿ ,G(X) a given vector withn real components defined in ? ^u ,U an unknown vector withs real components,K a known vector withs real components andA a given reals×n matrix of ranks. Assuming that, for every pair of pointsX ₁ , X₂of ? ⁿ ,G(X) satisfies the conditions $$(G(X_1 ) - G(X_2 ), X_1 - X_2 ) \geqslant o (X_1 - X_2 , X_1 - X_2 )$$ and $$\left\| {(G(X_1 ) - G(X_2 )\left\| { \leqslant M} \right\|X_1 - X_2 )} \right\|$$ wherec andM are positive constants, we prove that a unique solution of the system $$\left\{ \begin{gathered} G(X) + A ^T U = 0 \hfill \\ AX = K \hfill \\ \end{gathered} \right.$$ exists and we show a method for finding such a solution     

Generalizzazione di un metodo alle differenze finite per una classe di problemi al contorno di tipo ellittico

In a previous paper the numerical solution of a particular boundary-value problem for the «weakly linear» equation $$\Delta \left[ {u(P)} \right] = f(P,u)$$ was obtained and the convergence of a suitable finite-difference scheme was proved. This paper is concerned with the more general equation $$L\left[ {u(P)} \right] = f(P,u)$$ where $$L \equiv - \left[ {a\frac{\partial }{{\partial x^2 }} + c\frac{\partial }{{\partial y^2 }} + d\frac{\partial }{{\partial x}} + e\frac{\partial }{{\partial y}}} \right]$$ ; the solution is obtained using the same finite-difference scheme as in the previous paper, and sufficient condition for its convergence are given for this new case     

Numerical evaluation of a solution of a special mixed-type differential-difference equation

The function * $$f(t) = \frac{{e^{ - \alpha \gamma } }}{\pi }\int\limits_0^\infty {\cos t \xi e^{\alpha Ci(\xi )} \frac{{d\xi }}{{\xi ^\alpha }},t \in R,\alpha > 0} $$ [Ci(x)=cosine integral, γ=Euler's constant] is studied and numerically evaluated;f is a solution to the following mixed type differential-difference equation arising in applied probability: ** $$tf'(t) = (\alpha - 1)f(t) - \frac{\alpha }{2}[f(t - 1) + f(t + 1)]$$ satisfying the conditions: i) $$f(t) \geqslant 0,t \in R$$ , ii) $$f(t) = f( - t),t \in R$$ , iii) $$\int\limits_{ - \infty }^{ + \infty } {f(\xi )d\xi = 1} $$ . Besides the direct numerical evaluation of (*) and the derivation of the asymptotic behaviour off(t) fort→0 andt→∞, two different iterative procedures for the solution of (**) under the conditions (i) to (iii) are considered and their results are compared with the corresponding values in (*). Finally a Monte Carlo method to evaluatef(t) is considered.     

On the computation of Tricomi's ψ function

Two methods for calculating Tricomi's confluent hypergeometric function are discussed. Both methods are based on recurrence relations. The first method converges like $$\exp ( - \alpha |\lambda |^{1/3} n^{2/3} )for some \alpha > 0$$ and the second like $$\exp ( - \beta |\lambda |^{1/2} n^{1/2} )for some \beta > 0.$$ Several examples are presented.     

Resolution numerique de quelques problemes raides en mecanique des milieux faiblement compressibles

This paper deals with the numerical approximation of some «stiff» problems by asymptotic expansion of the solution. The model problem is the stationary linearized equation of slightly compressible fluids: $$\begin{gathered} \ll Find u_\varepsilon \varepsilon \left[ {H_0^1 (\Omega )} \right]^n s.t. \hfill \\ - \mu \Delta u_\varepsilon - \frac{1}{\varepsilon } grad div u_\varepsilon = f in \Omega \gg \hfill \\ \end{gathered} $$ where ∈ is asmall parameter; numerical treatment of the problem is thus difficult («stiff» problem). We establish existence and unicity of an asymptotic expansion foru, and use it to computeu. In the usual cases, with small divergence, the numerical results are far better than those obtained by direct discretisation of the problem. We also construct asymptotic expansions for the solutions of some nonlinear or non-stationary related problems.     

Un metodo del terzo ordine per l'integrazione numerica dell'equazione differenziale ordinaria

For the numerical integration of the ordinary differential equation $$\frac{{dy}}{{dx}} = F(x,y) y(x_0 ) = y_0 \begin{array}{*{20}c} x \\ {x_0 } \\ \end{array} \varepsilon [a,b]$$ a third method utilizing only two points for every step, is determined different from the analogous Runge-Kutta method employing three points; it is useless take the first step as the «pseudo Runge-Kutta method». The truncation error is given, the convergence is proved and finally a numerical exercise is given.     

Alcuni problemi relativi alle formule di quadratura del tipo di tchebycheff

In this paper we study quadrature formulas of the form $$\int\limits_{ - 1}^1 {(1 - x)^a (1 + x)^\beta f(x)dx = \sum\limits_{i = 0}^{r - 1} {[A_i f^{(i)} ( - 1) + B_i f^{(i)} (1)] + K_n (\alpha ,\beta ;r)\sum\limits_{i = 1}^n {f(x_{n,i} ),} } } $$ (α>?1, β>?1), with realA _i ,B _i ,K _n and real nodesx _n,i in (?1,1), valid for prolynomials of degree ≤2n+2r?1. In the first part we prove that there is validity for polynomials exactly of degree2n+2r?1 if and only if α=β=?1/2 andr=0 orr=1. In the second part we consider the problem of the existence of the formula $$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} f(x)dx = A_n f( - 1) + B_n f(1) + C\sum\limits_{i = 1}^n {f(x_{n,i} )} }$$ for polynomials of degree ≤n+2. Some numerical results are given when λ=1/2.     

A note on the Bernstein algorithm for bounds for interval polynomials

In computing the range of values of a polynomial over an intervala≤x≤b one may use polynomials of the form $$\left( {\begin{array}{*{20}c} k \\ j \\ \end{array} } \right)\left( {x - a} \right)^j \left( {b - x} \right)^{k - j} $$ called Bernstein polynomials of the degreek. An arbitrary polynomial of degreen may be written as a linear combination of Bernstein polynomials of degreek≥n. The coefficients of this linear combination furnish an upper/lower bound for the range of the polynomial. In this paper a finite differencelike scheme is investigated for this computation. The scheme is then generalized to interval polynomials.