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Symmetry preserving numerical schemes for partial differential equations and their numerical tests
 J. Difference Equ. Appl
"... The method of equivariant moving frames on multispace is used to construct symmetry preserving finite difference schemes of partial differential equations invariant under finitedimensional symmetry groups. Invariant numerical schemes for a heat equation with a logarithmic source and the spherical ..."
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The method of equivariant moving frames on multispace is used to construct symmetry preserving finite difference schemes of partial differential equations invariant under finitedimensional symmetry groups. Invariant numerical schemes for a heat equation with a logarithmic source and the spherical Burgers equation are obtained. Numerical tests show how invariant schemes can be more accurate than standard discretizations on uniform rectangular meshes. 1
Lie group computation of finite difference schemes
, 2006
"... Abstract. A Mathematica based program has been elaborated in order to determine the symmetry group of a finite difference equation. The package provides functions which enable us to solve the determining equations of the related Lie group. 1 ..."
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Abstract. A Mathematica based program has been elaborated in order to determine the symmetry group of a finite difference equation. The package provides functions which enable us to solve the determining equations of the related Lie group. 1
Invariant discretization schemes using evolution–projection techniques
"... Abstract. Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the onedimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for noninvariant schemes of the heat equation in com ..."
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Abstract. Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the onedimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for noninvariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution–projection strategy. Key words: invariant numerical schemes; moving frame; evolution–projection method; heat equation 2010 Mathematics Subject Classification: 65M06; 58J70; 35K05 1
Symmetry, Integrability and Geometry: Methods and Applications Invariant Discretization Schemes Using Evolution–Projection Techniques ⋆
"... Abstract. Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the onedimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for noninvariant schemes of the heat equation in comp ..."
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Abstract. Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the onedimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for noninvariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution–projection strategy. Key words: invariant numerical schemes; moving frame; evolution–projection method; heat equation 2010 Mathematics Subject Classification: 65M06; 58J70; 35K05 1
unknown title
, 2009
"... Multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test. I ..."
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Multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test. I
Manuscript submitted to Website: www.aimSciences.org DCDS Supplement Volume 2007 pp. 1–10 LIE GROUP STABILITY OF FINITE DIFFERENCE SCHEMES
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