Adaptive Energy Preserving Methods for Partial Differential Equations
For computational scientists solving PDEs, this provides a way to maintain conservation laws under adaptive mesh refinement, though it is an incremental extension of existing energy-preserving techniques.
The paper presents a method for constructing energy-preserving numerical schemes for time-dependent PDEs on non-uniform grids, extending to r-, h-, and p-adaptivity. Applied to KdV and Sine-Gordon equations, numerical results demonstrate the method's effectiveness.
A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used with both finite difference and partition of unity approaches, thereby also including finite element approaches. The schemes are then extended to accommodate $r$-, $h$- and $p$-adaptivity. The method is applied to the Korteweg-de Vries equation and the Sine-Gordon equation and results from numerical experiments are presented.