Difference schemes with point symmetries and their numerical tests

Abstract

Abstract. Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new difference schemes are tested as numerical methods. The obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost. For an example involving a solution with a singularity in the integration region the symmetry preserving scheme, contrary to standard ones, provides solutions valid beyond the singular point.

Lie-point symmetries of the discrete Liouville equation

“…00 u 01 u 10 (x 10 − x 00 )(y 01 − y 00 ) + (6.7) +Q (3) 10 u 01 u 00 (x 10 − x 00 )(y 01 − y 00 ) + Q (3) 01 u 00 u 10 (x 10 − x 00 )(y 01 − y 00 ) + +u 00 u 01 u 10 (Q (1) 10 − Q (1) 00 )(y 01 − y 00 ) + u 00 u 01 u 10 (x 10 − x 00 )(Q (2) 01 − Q (2) 00 ).…”

Symmetry-Preserving Numerical Schemes

“…The KdV equation then reduces to 15) together with the companion equations (2.11), (2.14). In particular, when k = 1 in (2.12), we obtain the system of differential equations…”

“…For ordinary differential equations, symmetry-preserving schemes have proven to be very promising. For solutions exhibiting sharp variations or singularities, symmetry-preserving schemes systematically appear to outperform standard numerical schemes, [15,16,20,53]. For partial differential equations, the improvement of symmetry-preserving schemes versus traditional integrators is not as clear, [7,52,56,78].…”

Transformation groups on real plane and their differential invariants

“…Such realizations are also applicable in the difference schemes for numerical solutions of differential equations [4]. Description of realizations is the first step for solving the Levine's problem [17] on the second-order time-independent Hamiltonian operators which lie in the universal enveloping algebra of a finite-dimensional Lie algebra of the first-order differential operators.…”