Method of Moving Frames to Solve Conservation Laws on Curved Surfaces

A new numerical framework is proposed to solve partial differential equations on curved surfaces by using the orthogonal moving frames at each grid point to compute the gradient of a scalar variable. We call this framework the method of moving frames (MMF) that is adopted and modified from the works of É. Cartan. Compared to the Eulerian method and the Lagrangian multiplier method, the MMF method uses only the surface as the domain, not additionally the ambient space enclosing it. Also different from directly solving the equations with respect to the curved axis, the MMF method is free of the metric tensors. This uniqueness is the consequence of the virtual and penalty extension of the variables in a special direction, called the exponential direction, instead of the surface normal direction that is typically taken. The exponential extension eliminates the need to extend the computational domain and the variables outside the curved surfaces, but the variables outside the curved surfaces are not extended in the direction of the surface normal, yielding an extension error. However, the overall error for the MMF scheme, caused by the extension error, is of high order in L2 error with respect to space discretization. This high convergence rate implies that the exponential error can be made negligible compared to the error of differentiation and integration, which are also expressed with space discretization but with lower order, in adaptively-refined meshes proportional to the Gaussian curvature. As the first application of the MMF method, conservation laws are considered on curved surfaces. To display the exponential convergence and the unique features of the MMF scheme, convergence tests are demonstrated on four different types of surfaces: an open spherical shell, a closed spherical shell, an irregular closed surface, and a non-convex closed surface.