This review paper examines a synthesis of adaptive mesh methods with the use of symmetry to solve ordinary and partial differential equations. It looks at the effectiveness of numerical methods in preserving geometric structures of the underlying equations such as scaling invariance, conservation laws and solution orderings. Studies are made of a series of examples including the porous medium equation and the nonlinear Schrödinger equation.
Budd, C. J., & Piggott, M. D. (2001). The geometric integration of scale-invariant ordinary and partial differential equations. Journal of Computational and Applied Mathematics, 128(1-2), 399-422. https://doi.org/10.1016/S0377-0427(00)00521-5