Algorithmic Approach to Strong Consistency Analysis of Finite Difference Approximations to PDE Systems

Gerdt, Vladimir P.; Robertz, Daniel

doi:10.1145/3326229.3326255

Cited by 8 publications

(11 citation statements)

References 20 publications

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“…[53], p. 50) to Eqs. (25) and to (24) as well. Clearly, the differential system (25), ( 27) is passive and simple (cf.…”

Section: One Can Rewrite System (23) Equivalently Asmentioning

confidence: 80%

“…The generalization to nonlinear PDE given in [18], based on the concept of difference Gröbner basis, is not algorithmic, because the difference polynomial ring is non-Noetherian [22,36] and the basis may be infinite. In the conference paper [25] we suggested an algorithm for verification of s-consistency on Cartesian grids that is based on investigating the FDA by a difference analogue of differential Thomas decomposition. In the special case of binomial perfect difference ideals another kind of decomposition was suggested in [16].…”

Section: Introductionmentioning

confidence: 99%

“…However, the concept of s-consistency, as it was introduced in [23,18,25], is applicable to Cartesian grids only. Its generalization to more general regular grids, whose grid spacings may be pairwise different, requires certain modifications and extensions.…”

Section: Introductionmentioning

confidence: 99%

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Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations

Gerdt,

Robertz,

Blinkov

2020

Preprint

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For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach that combines differential and difference algebra to analyze s(trong)-consistency of finite difference approximations. Our approach is applicable to regular solution grids. For the grids of this type we give a new definition of s-consistency for finite difference approximations which generalizes our definition given earlier for Cartesian grids. The algorithmic verification of s-consistency presented in the paper is based on the use of both differential and difference Thomas decomposition. First, we apply the differential decomposition to the input system, resulting in a partition of its solution space. Then, to the output subsystem that contains a solution of interest we apply a difference analogue of the differential Thomas decomposition which allows to check the s-consistency. For linear and some quasi-linear differential systems one can also apply difference Gröbner bases for the s-consistency analysis. We illustrate our methods and algorithms by a number of examples, which include Navier-Stokes equations for viscous incompressible flow.

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“…[53], p. 50) to Eqs. (25) and to (24) as well. Clearly, the differential system (25), ( 27) is passive and simple (cf.…”

Section: One Can Rewrite System (23) Equivalently Asmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations

Gerdt,

Robertz,

Blinkov

2020

Preprint

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“…The FDM has a simple form, exibility (Yan et al (2016); Mori and Romao (2015)). To build numerical solutions, it is necessary to carry out the proper discretization as initial conditions and boundary conditions that can provide approximate solutions for surface wave motion (Gerdt and Robertz, 2019). The main advantage of FDM is that it is an asymptotic algorithm that can simulate an entire wave eld without losing accuracy.…”

Section: Introductionmentioning

confidence: 99%

Surface Wave Topography using The 4 Point FDM Simulator

Jufriansah¹,

Pramudya²,

Hermanto³

et al. 2020

sci. technol. indones.

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The 2D topography proffers a new challenge of modeling surface waves with a 4-point finite difference (FDM) model. Topographic representation of wave propagation over a certain area will result in loss of accuracy of the numerical model. Then from this the need for appropriate modifications to reduce calculation errors. The existing approach requires value representation as an internal extrapolation solution for temporary exterior conditions. It is finally by providing boundary conditions and initial conditions in the system. However, the scheme sometimes becomes unstable for very irregular topography. 1D extrapolation along the parallel path is known to produce a simple and efficient scheme. During extrapolation, the stability of the 1D hyperbolic Schema improved by disregarding the nearest interior boundary point, which is less than half the lattice distance. Given the limited difference so that the stencils from both sides of the central evaluation point can be used as a 2D form modification if there are not enough inside points. So that in propagation space, waves that move and change according to changes in time. It will be following the wave nature of one source that travels in the x and y fields whose amplitude will change exponentially against propagation time. It is by the nature of surface wave motion.

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“…We believe that our method can also be applied to obtain bounds for other algorithms in differential algebra such as [1,Algorithm 3.6] and for algorithms from other theories, e.g. [7,Algorithm 3] for systems of difference equations. Since the reducibility of a polynomial can be expressed as a first-order existential formula, it seems plausible that the same methods could be applied to other algorithms dealing with difference [5] and differential-difference [6] equations that use factorization because the corresponding theories satisfy the requirements of our approach [14,17,23].…”

Section: Introductionmentioning

confidence: 99%

Algorithms yield upper bounds in differential algebra

Li¹,

Ovchinnikov²,

Pogudin³

et al. 2020

Preprint

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Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the input. We also generalize this to algorithms working with models of good enough theories (including for example, difference fields).We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.

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Algorithmic Approach to Strong Consistency Analysis of Finite Difference Approximations to PDE Systems

Cited by 8 publications

References 20 publications

Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations

Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations

Surface Wave Topography using The 4 Point FDM Simulator

Algorithms yield upper bounds in differential algebra

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