No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
A Calculus of Difference Schemes for the Solution of Boundary Value Problems on Irregular Meshes
Abstract
This paper derives compact-as-possible difference schemes for the solution of high-order two-point boundary-value problems on irregular meshes. This is accomplished by first writing the high-order equation as a first-order system of equations, discretizing and then algebraically reducing the discrete system to a compact-as-possible difference scheme for the original high-order equation. The reduced scheme inherits the properties of the discrete first-order system. In particular, if the first-order system represents a centered Euler scheme, then the reduced scheme will be second-order accurate despite possibly inconsistent truncation error. This phenomenon is known as supraconvergence.
... Actually consistency is not necessary, the scheme maintains the accuracy and the global error behaves better than the local error would indicate. This property of enhancement of the truncation error is called supra-convergence and for second and higher order boundary value problems, this phenomenon, discovered by Tikhonov and Samarskij [29], was widely analyzed in various cases by Manteuffel and his co-authors in [20], [21], [16], [22]. ...
In this paper we estimate the error of upwind first order finite volume schemes applied to scalar conservation laws. As a first step, we consider standard upwind and flux finite volume scheme discretization of a linear equation with space variable coefficients in conservation form. We prove that, in spite of their lack of consistency, both schemes lead to a first order error estimate. As a final step, we prove a similar estimate for the nonlinear case. Our proofs rely on the notion of geometric corrector, introduced in our previous paper by Bouche et al. (2005) [24] in the context of constant coefficient linear advection equations.
... Actually consistency is not necessary, the scheme maintains the accuracy and the global error behaves better than the local error would indicate. This property of enhancement of the truncation error is called supra-convergence and for second and higher order boundary value problems, this phenomenon discovered by Tikhonov and Samarskij [26] was widely analyzed in various cases by Manteuffel and his co-authors in [19], [20], [15], [21] and in Garcia-Archilla and Sanz-Serna [11]. ...
This paper deals with the upwind nite volume method applied to the linear advec- tion equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector which is a sequence associated with every nite volume mesh in Rnd and every non vanishing vector a of Rnd. First we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size then an order one error estimate for the nite volume scheme occurs. Then we prove that this norm is indeed bounded by the mesh size in various cases including the one where an arbitrary coarse conformal triangular mesh is uniformly rened in two dimension. Computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases like the one with independent rened meshes.
This paper investigates the order of convergence of the upwind nite volume method for solving linear steady convection problem on a bounded domain, with natural boundary conditions. In order to overcome the nonconsistency in the nite differences sense, we intro- duce what we call the geometric correctors, which is associated with every nite volume mesh and every constant convection vector. Under a local quasi-uniformity condition for the trian- gulation and if the continuous solution is regular enough, we r st establish a link between the convergence of the nite volume scheme and these geometric correctors. Hence the study of this corrector in the case of uniformly rened,triangular meshes leads to the proof of the optimal or- der of convergence. We then focus our attention on the Peterson case where for a nonuniformly rened mesh and a convection direction parallel to one side of the domain, a loss of accuracy was proved. R…SUM…. Nous nous intØressons ‡ l’ordre de convergence du schØma volumes,nis,appliquØ ‡ l’Øquation d’advection linØaire stationnaire dans un domaine,bornØ. An de contourner la non consistance au sens des diffØrences nies, nous introduisons un correcteur gØomØtrique ne dØpendant que du maillage et du vecteur de convection. Sous une hypothse de quasi-uniformitØ, nous dØmontrons que l’Øtude de la convergence du schØma volumes,nis,peut se ramener,‡ l’Øtude de ce correcteur si la solution est sufsamment,rØgulir e. Dans le cas d’un maillage de triangles uniformØment rafnØs, on dØmontre alors que la convergence est d’ordre 1. Ensuite, nous considØrons le cas test de Peterson, cas oø lorsque le maillage est rafnØ de faÁon non uniforme et l’advection parallle au bord, on observe une perte dans le taux de convergence. KEYWORDS: Finite Volume Method ; Convergence ; Geometric Corrector ; Unstructured Meshes. MOTS-CL…S : Volumes nis,; Convergence ; Correcteur gØomØtrique ; maillage non structurØ. 2,Finite volumes,for complex applications IV.
The purpose of this paper is to show that the cell-centered upwind Finite Volume scheme applied to general hyperbolic systems of m conservation laws approximates smooth solutions to the continuous problem at order one in space and time. As it is now well understood, there is a lack of consistency for order one upwind Finite Volume schemes: the truncation error does not tend to zero as the time step and the grid size tend to zero. Here, following our previous papers on scalar equations, we construct a corrector that allows us to prove the expected error estimate for nonlinear systems of equations in one dimension.
The system of equations for the analysis of rotationally symmetric shells under time-dependent or static surface loadings has been formulated with the transverse, meridional, and circumferential displacements as the dependent variables. All loading functions must be continuous. The thickness h of the shell may vary along the meridian. Four of the eight natural boundary conditions will be prescribed as time-dependent boundary conditions at each boundary edge of the shell. Surface loadings and inertia forces in the three displacement directions of the shell have been considered. Fourier series are used in the circumferential direction of the shell. Solutions for each Fourier component are found by replacing all derivatives by their finite difference equivalents and solving the resulting system of algebraic equations at successive increments of the time variable. The complete system of equations is solved implicitly for the first time increment, while explicit relations are used to determine the three primary displacements within the boundary edges of the shell for the second and succeeding time increments. The remaining unspecified primary variables are then determined by separate implicit solutions at each boundary for the second and succeeding time increments. Subsequently, all remaining primary and secondary variables are found explicitly. A variable node point spacing may be specified over the full range of the spatial finite difference mesh. A numerical stability criterion which will ensure stable solutions for selected time increments and spatial meshes has been derived and found to agree with results for typical example solutions. Numerical solutions obtained with the computer program which accompanies this development have been found to be stable and in agreement for a wide range of practical values of both spatial and time increments for typical shells and loadings. Static and dynamic solutions for a parabolic shell with fixed boundary conditions at one edge and free boundary conditions at the other edge have been obtained using both constant and variable node point spacings and presented as examples.
In this paper we give a detailed analysis of the central difference scheme applied to the linear second-order equation ɛx″ = a(t)x′ + b(t)x + ƒ(t), with two-point boundary conditions x(t0) = x0, x(tf) = xf. We assume that a(t) < 0, so the solution x(t) may have a boundary layer in t0. A class of nonuniform meshes is proposed, which ensures accuracy in the layer plus stability and second-order convergence elsewhere.
This work addresses a theory of convergence for finite volume methods applied to linear equations. A non-consistent model problem posed in an abstract Banach space is proved to be convergent. Then various examples show that the functional framework is non-empty. Convergence with a rate h 1/2 of all TVD schemes for linear advection in 1D is an application of the general result. Using duality techniques and assuming enough regularity of the solution, convergence of the upwind finite volume scheme for linear advection on a 2D triangular mesh is proved in Lα, 2 ≤ α ≤ +∞: provided the solution is in W1,∞, it proves a rate of convergence h1/4-ε in L∞.
A general theory of difference methods for problems of the form
Ny ≡ y' - f(t,y) = O, a ≦ t ≦ b, g(y(a),y(b))= 0,
is developed. On nonuniform nets, t_0 = a, t_j = t_(j-1) + h_j, 1 ≦ j ≦ J, t_J = b, schemes of the form
N_(h)u_j = G_j(u_0,•••,u_J) = 0, 1 ≦ j ≦ J, g(u_0,u_J) = 0
are considered. For linear problems with unique solutions, it is shown that the difference scheme is stable and consistent for the boundary value problem if and only if, upon replacing the boundary conditions by an initial condition, the resulting scheme is stable and consistent for the initial value problem. For isolated solutions of the nonlinear problem, it is shown that the difference scheme has a unique solution converging to the exact solution if (i) the linearized difference equations are stable and consistent for the linearized initial value problem, (ii) the linearized difference operator is Lipschitz continuous, (iii) the nonlinear difference equations are consistent with the nonlinear differential
equation. Newton’s method is shown to be valid, with quadratic convergence, for computing the numerical solution.
We consider the general system of n first order linear
ordinary differential equations y'(t)=A(t)y(t)+g(t), a<t< b,
subject to "boundary" conditions, or rather linear constraints, of the form Σ^(N)_(ν=1) B_(ν)y(τ_ν)=β
Here y(t), g(t) and II are n-vectors and A(t), Bx,..., BN are n × n matrices. The N distinct points {τ_ν} lie in [a,b] and we only require N ≧ 1. Thus as special cases
initial value problems, N=1, are included as well as the general 2-point boundary value problem, N=2, with τ_1=a, τ_2=b. (More general linear constraints are also studied, see (5.1) and (5.17).)
The boundary value problem for ordinary differential equations is considered and a general theory for difference approximation is developed. In particular, the influence of extra boundary conditions is investigated and the eigenvalue problem is considered in detail.
An explicit reduction is given of the centered Euler scheme for a first order two-dimensional system producing an accurate difference approximation for a second order scalar equation. All of the advantages of the centered Euler formula are retained except for the condition number. Results of numerical experiments are presented.
In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes. A technique of reducing centered-difference approximations of first-order systems to discretizations of the underlying scalar equation is developed. We treat both vertex-centered and cell-centered difference schemes and indicate how these results apply to partial differential equations on Cartesian product grids.
As Tikhonov and Samarskii-breve showed for k = 2, it is not essential that kth-order compact difference schemes be centered at the arithmetic mean of the stencil's points to yield second-order convergence (although it does suffice). For stable schemes and even k, the main point is seen when the kth difference quotient is set equal to the value of the kth derivative at the middle point of the stencil; the proof is particularly transparent for k = 2. For any k, in fact, there is a (k/2)-parameter family of symmetric averages of the values of the kth derivative at the points of the stencil which, when similarly used, yield second-order convergence. The result extends to stable compact schemes for equations with lower-order terms under general boundary conditions. Although the extension of Numerov's tridiagonal scheme (approximating D/sup 2/y = f with third-order truncation error) yields fourth-order convergence on meshes consisting of bounded number of pieces in which the mesh size changes monotonically, it yields only third-order convergence to quintic polynomials on any three-periodic mesh with unequal adjacent mesh sizes and fixes adjacent mesh ratios. A result of some independent interest is appended (and applied): it characterizes, simply, those functions of k variables which possess the property that their average value, as one translates over one period of an arbitrary periodic sequence of arguments, is zero; i.e., those bounded functions whose average value, as one translates over arbitrary finite sequences of arguments, goes to zero as the length of the sequences increases.
For j k-2 (B.11b) For j < k-2 (B.11c) Plugging (B.11a, b, c) into (B.9) yields the result
- Thomas A Manteuffel And Andrew B
- Jr White
THOMAS A. MANTEUFFEL AND ANDREW B. WHITE, JR. For j k-2 (B.11b) For j < k-2 (B.11c) Plugging (B.11a, b, c) into (B.9) yields the result. REFERENCES