On the effects of lifting on interpolating wavelets for combustion simulations

The use of wavelets - and more generally multi-resolution analysis (MRA) - for the solution of non-linear partial differential equations (PDEs) is an active area of research, with many schemes currently available. Recently, a scheme has been developed, which employs biorthogonal interpolating wavelets, and which has been applied to problems in combustion [31]. Of central importance to this method is the discretization of the derivatives appearing in the governing equations. In reference [31], the derivative approximations are expressed in terms of an assembly of submatrices, each of which describes the interactions of wavelets and their derivatives across a range of scales and spatial locations. In the current paper, the accuracy and stability of derivative approximations based on interpolating wavelets are also found. Scaling functions with a large polynomial span are constrained to provide numerical approximations with a formal accuracy less than the maximum attainable for a given computational stencil. It is also found that approximations based on wavelets with a large polynomial span are unstable for non-periodic discretizations. In addition, lifting the MRA does little to improve the situation. The influence of lifting on the numerical implementation and behaviour of our numerical scheme is examined, and found to be comparatively minor.