Refinement of SOR method for the rational finite difference solution of first-order Fredholm integro-differential equations

Изучены монотонность и точность модифицированной схемы КАБАРЕ, аппроксимирующей квазилинейную гиперболическую систему законов сохранения. Получены условия, при которых эта схема сохраняет монотонность разностного решения относительно инвариантов линейного приближения аппроксимируемой системы. В качестве конкретного примера рассмотрена аппроксимация системы законов сохранения теории мелкой воды. На примере этой системы показано, что подобно TVD-схемам повышенного порядка аппроксимации на гладких решениях схема КАБАРЕ, несмотря на высокую точность при локализации ударных волн, снижает свой порядок сходимости в областях их влияния. We studied the monotonicity and the accuracy of the modified CABARET scheme approximating the quasilinear hyperbolic system of conservation laws. We obtained conditions under which this scheme preserves the monotonicity of the finite-difference solution with respect to invariants of the linear approximation of the considering system. As a specific example, we considered the approximation for the system of conservation laws for shallow water theory. To determine the accuracy of the scheme in the regions influenced by shocks that propagate with variable velocity and behind the fronts of which a non-constant solution is formed, series of test calculations on the sequence of contracting grids were carried out. These calculations enabled us to apply the Runge rule to define the order of convergence of the finite-difference solution. To estimate the accuracy of the modified CABARET scheme, the orders of its integral and local convergence are calculated, as well as local disbalances in calculating of the absolute value of averaged basis functions vector and local disbalances in calculating of the absolute value of the components of the averaged vector of invariants. The order of the integral convergence makes it possible to estimate the accuracy of finite-difference schemes in translation of the RankineHugoniot conditions across a non-stationary shock wave front. In this case, the order of integral convergence is calculated for the finite-difference solution itself (rather than for its absolute value, as in the L1 norm) on spatial intervals crossing the shock wave. It is shown that, like TVD-schemes of high order approximation on smooth solutions, the CABARET scheme being formally of the second order, in spite of high accuracy in the localization of shocks, reduces the order of integral convergence to the first order on the integration intervals, one of the boundaries of which is in the region influenced by a shock. The main reason of this decrease in accuracy is that the flux correction used in the CABARET scheme to its monotonization leads to the decrease in the smoothness of the finite-difference fluxes, which in turn leads to the decrease in the order for approximation of Rankine-Hugoniot conditions on the shocks. As a result, the local accuracy of the CABARET scheme in the regions of shocks influence also decreases to about the first order.

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ERRORS IN THE FINITE DIFFERENCE SOLUTION OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

Control of Distributed Parameter Systems ◽

10.1016/b978-0-08-022018-5.50024-0 ◽

1978 ◽

pp. 193-200

Author(s):

D. Maudsley

Keyword(s):

Partial Differential Equations ◽

Finite Difference ◽

Differential Equations ◽

Hyperbolic Partial Differential Equations ◽

Partial Differential ◽

Finite Difference Solution ◽

Difference Solution

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Effect of Plastic Anisotropy on the Distribution of Residual Stresses and Strains in Rotating Annular Disks

Symmetry ◽

10.3390/sym10090420 ◽

2018 ◽

Vol 10 (9) ◽

pp. 420 ◽

Cited By ~ 3

Author(s):

Woncheol Jeong ◽

Sergei Alexandrov ◽

Lihui Lang

Keyword(s):

Finite Difference ◽

Differential Equations ◽

Residual Stresses ◽

Boundary Value Problem ◽

Boundary Value ◽

Plastic Anisotropy ◽

Strain Rates ◽

Finite Difference Solution ◽

Residual Stresses And Strains ◽

Difference Solution

Hill’s quadratic orthotropic yield criterion is used for revealing the effect of plastic anisotropy on the distribution of stresses and strains within rotating annular polar orthotropic disks of constant thickness under plane stress. The associated flow rule is adopted for connecting the stresses and strain rates. Assuming that unloading is purely elastic, the distribution of residual stresses and strains is determined as well. The solution for strain rates reduces to one nonlinear ordinary differential equation and two linear ordinary differential equations, even though the boundary value problem involves two independent variables. The aforementioned differential equations can be solved one by one. This significantly simplifies the numerical treatment of the general boundary value problem and increases the accuracy of its solution. In particular, comparison with a finite difference solution is made. It is shown that the finite difference solution is not accurate enough for some applications.

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Time-Differencing Schemes

An Introduction to Global Spectral Modeling ◽

10.1093/oso/9780195094732.003.0005 ◽

1998 ◽

Author(s):

T. N. Krishnamurti ◽

H. S. Bedi ◽

V. M. Hardiker

Keyword(s):

Exact Solution ◽

Partial Differential Equations ◽

Finite Difference ◽

Differential Equations ◽

Higher Order ◽

Second Order ◽

Time Step ◽

Partial Differential ◽

Finite Difference Solution ◽

Difference Solution

Atmospheric models generally require the solutions of partial differential equations. In spectral models, the governing partial differential equations reduce to a set of coupled ordinary nonlinear differential equations where the dependent variables contain derivatives with respect to time as well. To march forward in time in numerical weather prediction, one needs to use a time-differencing scheme. Although much sophistication has emerged for the spatial derivatives (i.e., second- and fourth-order differencing), the time derivative has remained constructed mostly around the first- and second-order accurate schemes. Higher-order schemes in time require the specification of more than a single initial state, which has been considered to be rather cumbersome. Therefore, following the current state of the art, we focus on the first- and second-order accurate schemes. However, higher-order schemes, especially for long-term integrations such as climate modeling, deserve examination. We start with the differential equation dF/dt = G, where F = F(t) and G = G(t). If the exact solution of the above equation can be expressed by trigonometric functions, then our problem would be to choose an appropriate time step in order to obtain a solution which behaves properly; that is, it remains bounded with time. This is illustrated in Fig. 3.1. We next show that: (1) if an improper time step is chosen, then the approximate finite difference solution may become unbounded, and (2) if a proper time step is chosen, then the finite difference solution will behave quite similar to the exact solution. The stability or instability of a numerical scheme will be discussed for a single Fourier wave. This would also be valid for a somewhat more general case, since the total solution is a linear combination of sine and cosine functions, which are all bounded. We next define an amplification factor |λ|, the magnitude of which would determine whether a scheme is stable or not.

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