Abstract:

Numerically stable formulas are presented for the closedform analytical solution of the XIVAS scheme in 3D. This scheme is a stateoftheart particlebased explicit exponential integrator developed for the particle finite element method. Algebraically, this scheme involves two steps: (1) the solution of tangent curves for piecewise linear vector fields defined on simplicial meshes and (2) the solution of line integrals of piecewise linear vectorvalued functions along these tangent curves. Hence, the stable formulas presented here have general applicability, e.g. exact integration of trajectories in particlebased (Lagrangiantype) methods, flow visualization and computer graphics. The Newton form of the polynomial interpolation definition is used to express exponential functions of matrices which appear in the analytical solution of the XIVAS scheme. The divided difference coefficients in these expressions are defined in a piecewise manner, i.e. in a prescribed neighbourhood of removable singularities their series approximations are computed. An optimal series approximation of divided differences is presented which plays a critical role in this methodology. At least ten significant decimal digits in the formula computations are guaranteed to be exact using doubleprecision floatingpoint arithmetic. The worst case scenarios occur in the neighbourhood of removable singularities found in fourthorder divided differences of the exponential function. 