Lie Groups and Numerical Solutions of Differential Equations: Invariant Discretization Versus Differential Approximation

Abstract

Abstract. We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which the "first differential approximation" is invariant under the same Lie group as the original ordinary differential equation.

The Korteweg–de Vries equation and its symmetry-preserving discretization

“…As was discussed in Section 3.6, the invariant Lagrangian schemes (32) and (33) using (34) are exact for the Galilean invariant solution (10). We verify this by numerically computing this solution and calculating the l ∞ -norm and the RMSE.…”

“…In order to complete the numerical scheme (32) and (33) it is necessary to formulate an equation for the grid velocity. In the purely Lagrangian scheme one uses the discretization of the relation (7), which is…”

“…The mesh is said to be equidistributing for ρ on a b The function ρ is called mesh density function or monitor function. For the practical implementation it is advantageous to convert the relation (36) into a differential equation. This is done by first using the equivalent expression where ξ j , = j N 1 ,..., , is the discrete computational coordinate.…”

“…For partial differential equations (PDEs) the first application of Lie group theory to numerical methods is, to our knowledge, due to Shokin and Yanenko [45,49]. Their approach "Differential approximation" is quite different from ours (for a comparison see [33]).…”

Lie groups and continuum mechanics: where do we stand today?

“…Another technique relies on the concept of differential approximations. It consists in finding a numerical scheme such that its differential approximation is invariant under the Lie group admitted by the equation [161][162][163]. It generally leads to only partially invariant schemes.…”