An error propagation in the numerical literature

Applied Mathematics and Computation, 2002

Numerical errors in solution of the Euler equations of fluid flow are studied. The error equations are solved to analyze the propagation of the discretization errors. In particular, the errors caused by the boundary conditions and their propagation are investigated. Errors generated at a wall are transported differently in subsonic and supersonic flows. Accuracy may be lost due to the accumulation of errors along the walls. This can be avoided by increasing the accuracy of the boundary conditions. Large errors may still arise locally at the leading edge of a wing profile. There, a fine grid is the best way to reduce the error.

ArXiv, 2020

The truncation and approximation errors for the set of numerical solutions computed by methods based on the algorithms of different structure are calculated and analyzed for the case of the two-dimensional steady inviscid compressible flow. The truncation errors are calculated using a special postprocessor, while the approximation errors are obtained by the comparison of the numerical solution and the analytic one. The extent of the independence of errors for the numerical solutions may be estimated via the Pearson correlation coefficient that may be geometrically expressed by the angle between errors. Due to this reason, the angles between the approximation errors are computed and related with the corresponding angles between the truncation errors. The angles between the approximation errors are found to be far from zero that enables a posteriori estimation of the error norm. The analysis of the distances between these solutions provides another approach to the estimation of the er...

Computational Methods in Applied Mathematics

Alexander Andreevich Samarskii (1919-2008) is one of the greats from the pleiad of Russian scientists who brought prominence, international recognition, and fame to the Soviet science. He is the founder of the mathematical modeling in Russia, the leading world expert in the field of computational mathematics, mathematical physics, theory of difference schemes, and numerical simulation of complex nonlinear systems. In 1999, A. A. Samarskii was awarded the State Prize for his work on the theory of difference schemes, [7]. Since receiving his Ph.D. in 1947 has held a series of appointments both at various universities and at the Russian Academy of Science. He has been Professor of M. V. Lomonosov Moscow State University and Keldysh Institute of Applied Mathematics of the Soviet Academy of Sciences for more than 50 years. In 1966, A. A. Samarskii was elected a corresponding member, and in 1976-a full member of the Academy of Sciences of the USSR. He is a Hero of Socialist Labor, Laureate of the Lenin and State Prizes and M. V. Lomonosov Prize, and awardee of numerous medals and orders as a veteran of the World War II. His early works were in the defense projects on the explosive power of nuclear, and later, of the hydrogen devices. A. A. Samarskii is the co-author of the scientific discovery "The effect of the T-layer", which is listed in the State Register of Discoveries of the USSR-the first recorded event detected first trough mathematical modeling and computer simulations, and only then in the real experiment. His cutting edge research in laser fusion, magnetic and radiation gas dynamics, high-power lasers, aerodynamics, nuclear energy, and plasma physics was the base for developing understanding of the principles and the organization of the concept of mathematical modeling and computational experiment. A. A. Samarskii created the general theory of operator-difference schemes, [8], including stability of difference schemes for elliptic, [9] and time-dependent problems, [10]. The main trust of Samarskii's scientific heritage is the theory of operator-difference schemes, [11]. After a proper discretization in space by finite differences or finite elements any initial value problem for time dependent linear partial differential equations (or systems) is transformed into Cauchy problem for a system of ordinary differential equations. This problem is set as an differential-operator equation in a proper finite dimensional Hilbert space. Time discretization leads to a operator-difference equation, which is studied by the developed by A. A. Samarskii theory of stability of such schemes. Fundamental result of Samarskii is the formulation of general and abstract criteria, namely, sufficient and necessary conditions, for the stability of two-and three-level schemes. These criteria are based on quite simple and easy to check operator inequalities. Using this approach, he developed the general theory of operator-difference schemes-the stability with respect to the initial data and the right-hand side and also stability with respect to problem coefficients (strong stability).

The extreme high-accuracy calculation of the 1D Poisson equation by the interpolation finite difference method (IFDM) is possible. Numerical calculation errors have conventionally been evaluated by comparison with theoretical solutions. However, it is not always possible to obtain a theoretical solution. In addition, when trying to obtain numerical values from the theoretical solution, the exact numerical value may not be obtained because of inherent difficulties, that is, a theoretical solution equal to the exact numerical solution does not hold. In this paper, we focus on an ordered structure of the error calculated by the high-order accuracy finite difference (FD) scheme. This approach clarifies that in the numerical calculation of the 1D Poisson equation, which is the most basic ordinary differential equation, the error of the numerical calculation can be evaluated without comparison with the theoretical solution. Furthermore, in the numerical calculation by the IFDM, not only t...

Journal of Computational and Applied Mathematics, 2000

This volume contains contributions in the area of di erential equations and integral equations. The editors wish to thank the numerous authors, referees, and fellow editors Claude Brezinski and Luc Wuytack, who have made this volume a possibility; it has been a major but personally rewarding e ort to compile it. Due to the limited number of pages we were obliged to make a selection when composing this volume. At an early stage it was agreed that, despite the connections between the subject areas, it would be beneÿcial to allocate the area of partial di erential equations to a volume for that area alone.

This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Editorial review

Advances in Difference Equations, 2013

In this paper, a numerical solution of partial differential-algebraic equations (PDAEs) is considered by multivariate Padé approximations. We applied this method to an example. First, PDAE has been converted to power series by two-dimensional differential transformation, and then the numerical solution of the equation was put into a multivariate Padé series form. Thus, we obtained the numerical solution of PDAEs.

2004

In this paper we present an estimate of accuracy for a piecewise polynomial approximation of a classical numerical solution to a non linear differential problem. We suppose the numerical solution u is computed using a grid with a small linear step and interval time Tu, while the polynomial approximation v is an interpolation of the values of a numerical solution on a less fine grid and interval time Tv << Tu. The estimate shows that the interpolant solution v can be, under suitable hypotheses, a good approximation and in general its computational cost is much lower of the cost of the fine numerical solution. We present two possible applications to linear case and periodic case.