Detecting Discontinuities Over Triangular Meshes Using Multiwavelets

It is well known that solutions to nonlinear hyperbolic PDEs develop discontinuities in time. The generation of spurious oscillations in such regions can be prevented by applying a limiter in the troubled zones. In earlier work, we constructed a multiwavelet troubled-cell indicator for one and (tensor-product) two dimensions (SIAM J. Sci. Comput. 38(1):A84–A104, 2016). In this paper, we investigate multiwavelet troubled-cell indicators on structured triangular meshes. One indicator uses a problem-dependent parameter; the other indicator is combined with outlier detection

Design of a Simple Orthogonal Multiwavelet Filter by Matrix Spectral Factorization

We consider the design of an orthogonal symmetric/antisymmetric multiwavelet from its matrix product filter by matrix spectral factorization (MSF). As a test problem, we construct a simple matrix product filter with desirable properties, and factor it using Bauer's method, which in this case can be done in closed form. The corresponding orthogonal multiwavelet function is derived using algebraic techniques which allow symmetry to be considered. This leads to the known orthogonal multiwavelet SA1, which can also be derived directly. We also give a lifting scheme for SA1, investigate the influence of the number of significant digits in the calculations, and show some numerical experiments.Comment: This is a preprint of a paper whose final and definite form is published in Circuits, Systems, and Signal Processing,, Springer, https://link.springer.com/article/10.1007/s00034-019-01240-9, ISSN 0278-081X (print), ISSN 1531-5878 (Online

Reviewed by Erwan Deriaz References

Non-uniform multiresolution analysis with supercompact multiwavelets. (English summary) J. Comput. Appl. Math. 235 (2010), no. 1, 334–340. This paper extends a previous paper on supercompact multiwavelets, by Beam and Warming, to the case of non-uniform partitions. The term supercompact multiwavelets means dyadic multiwavelets with the same minimal compact support as for the Haar wavelets. Multiwavelet bases {ψ1,..., ψℓ} result in the coexistence of different scale functions {ϕ1,..., ϕℓ} in one multiresolution analysis. In this article, the scale functions ϕi are the Legendre orthogonal polynomials. The two-scale relation is presented in its matricial formulation. Then passing to the non-uniform case induces the introduction of weights which modify the expressions issued from the uniform case