We introduce the concept of Hybrid Surrogate Models (HSMs) -- combining multivariate polynomials with Heavyside functions -- as approximates of functions with finitely many jump discontinuities. We exploit the HSMs for formulating a variational optimization approach, solving non-regular partial differential equations (PDEs) with non-continuous shock-type solutions. The HSM technique simultaneously obtains a parametrization of the position and the height of the shocks as well as the solution of the PDE. We show that the HSM technique circumvents the notorious Gibbs phenomenon, which limits the accuracy that classic numerical methods reach. Numerical experiments, addressing linear and non-linearly propagating shocks, demonstrate the strong approximation power of the HSM technique.
Recent developments in the context of minimum residual finite element methods are paving the way for designing finite element methods in non-standard function spaces. This, in particular, permits the selection of a solution space in which the best approximation of the solution has desirable properties. One of the biggest challenges in designing finite element methods are non-physical oscillations near thin layers and jump discontinuities. In this article we investigate Gibbs phenomena in the context of $L^q$-best approximation of discontinuities in finite element spaces with $1\leq q<\infty$. …
Adaptive rational interpolation has been designed in the context of image processing as a new nonlinear technique that avoids the Gibbs phenomenon when we approximate a discontinuous function. In this work, we present a generalization to the method giving explicit expressions for all the weights for any order of the algorithm. It has a similar behavior to weighted essentially non oscillatory (WENO) technique, however because of the design of the weights in this case is more simple, we propose a new way to construct them obtaining the maximum order near the discontinuities. Some experiments are…
One of the major challenges in finite element methods is the mitigation of spurious oscillations near sharp layers and discontinuities known as the Gibbs phenomenon. In this article, we propose a set of functionals to identify spurious oscillations in best approximation problems in finite element spaces. Subsequently, we adopt these functionals in the formulation of constraints in an effort to eliminate the Gibbs phenomenon. By enforcing these constraints in best approximation problems, we can entirely eliminate over- and undershoot in one dimensional continuous approximations, and significant…
A simple and very accurate method to approximate a function with a finite number of discontinuities is presented. This method relies on hyperbolic tangent functions of rational arguments as connecting functions at the discontinuities, each argument being the reciprocal of Newton binomials that depend on the abscissae that define the domain of the discontinuous function and upon the abscissae of discontinuities. Our approximants take the form of linear combinations of such hyperbolic tangent functions with coefficients that are obtained by solving a linear system of inhomogeneous equations whos…
A class of physics-informed spatio-temporal models has recently been proposed for modeling spatio-temporal processes governed by advection-diffusion equations. The central idea is to approximate the process by a truncated Fourier series and let the governing physics determine the dynamics of the spectral coefficients. However, because many spatio-temporal processes in real applications are non-periodic with boundary discontinuities, the well-known Gibbs phenomenon and ripple artifact almost always exist in the outputs generated by such models due to truncation of the Fourier series. Hence, the…
Data remap between non-matching meshes is a critical step in multiphysics coupling using a partitioned approach. The data fields being transferred often have jumps in function values or derivatives. It is important but very challenging to avoid spurious oscillations (a.k.a. the Gibbs Phenomenon) near discontinuities and at the same time to achieve high-order accuracy away from discontinuities. In this work, we introduce a new approach, called WLS-ENOR, or Weighted-Least-Squares-based Essentially Non-Oscillatory Remap, to address this challenge. Based on the WLS-ENO reconstruction technique pro…
In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial mapped bases allowing, for instance, to incorporate data or function discontinuities in a suitable mapping function. The new technique substantially mitigates the Runge's and Gibbs effects.
Pseudospectral schemes are a class of numerical methods capable of solving smooth problems with high accuracy thanks to their exponential convergence to the true solution. When applied to discontinuous problems, such as fluid shocks and material interfaces, due to the Gibbs phenomenon, pseudospectral solutions lose their superb convergence and suffer from spurious oscillations across the entire computational domain. Luckily, there exist theoretical remedies for these issues which have been successfully tested in practice for cases of well defined discontinuities. We focus on one piece of this …
Generalized Fourier Series: An N log2(N) extension for aperiodic functions that eliminates Gibbs oscillations
Khalifa University of Science and Technology；Emirates Nuclear Technology Center, Khalifa University of Science and Technology（阿联酋核技术中心，Khalifa University of Science and Technology）
Admissible Lax-Wendroff Flux Reconstruction Method with Automatic Differentiation on Adaptive Curved Meshes for Relativistic Hydrodynamics
Indian Institute of Technology Delhi（印度理工学院德里分校）；Johannes Gutenberg University Mainz（美因茨约翰内斯古腾堡大学）；IITD-Abu Dhabi Campus（德里印度理工学院阿布扎比分校）；Tata Institute of Fundamental Research（塔塔基础研究所）
A main disadvantage of many high-order methods for hyperbolic conservation laws lies in the famous Gibbs-Wilbraham phenomenon, once discontinuities appear in the solution. Due to the Gibbs-Wilbraham phenomenon, the numerical approximation will be polluted by spurious oscillations, which produce unphysical numerical solutions and might finally blow up the computation. In this work, we propose a new shock capturing procedure to stabilise high-order spectral element approximations. The procedure consists of going over from the original (polluted) approximation to a convex combination of the origi…
