We construct, on continuous $Q_1$ finite elements over Cartesian meshes, an interpolation operator that does not increase the total variation. The operator is stable in $L^1$ and exhibits second order approximation properties. With the help of it we provide improved error estimates for discrete minimizers of the total variation denoising problem and for total variation flows.
In this paper, we present a novel hybrid nonlinear explicit-compact scheme for shock-capturing based on a boundary variation diminishing (BVD) reconstruction. In our approach, we combine a non-dissipative sixth-order central compact interpolation and a fifth-order monotonicity preserving scheme (MP5) through the BVD algorithm. For a smooth solution, the BVD reconstruction chooses the highest order possible interpolation, which is central, i.e. non-dissipative in the current approach and for the discontinuities, the algorithm selects the monotone scheme. This method provides an alternative to t…
In this article, we offer a novel numerical approach for the solution of elastohydrodynamic lubrication line and point contact problems using a class of total variation diminishing (TVD) schemes on parallel computers. A direct parallel approach is presented by introducing a novel solver named as projected alternate quadrant interlocking factorization (PAQIF) by solving discrete variational inequality. For one-dimensional EHL case, we use weighted change in Newton-Raphson approximation to compute the Jacobian matrix in the form of a banded matrix by dividing two subregions on the whole computat…
Total-Variation-Diminishing Implicit-Explicit Runge-Kutta Methods for the Simulation of Double-Diffusive Convection in Astrophysics
University of Vienna, Faculty of Mathematics（维也纳大学数学系）；Universidad Pública de Navarra, Departamento de Ingeniería Matemática e Informática（纳瓦拉公立大学数学与计算机工程系）；Vienna University of Technology, Institute for Analysis and Scientific Computing（维也纳工业大学分析与科学计算研究所）
This paper presents the notion of a variation entropy. This concept is an entropy framework for the gradient of the solution of a conservation law instead of on the solution itself. It appears that all semi-norms are admissible variation entropies. This provides insight into the total variation diminishing property and justifies it from entropy principles. The framework allows to derive new local variation diminishing properties in the continuous form. This can facilitate the design of new numerical methods for problems that contain discontinuities.
Second order Implicit-Explicit Total Variation Diminishing schemes for the Euler system in the low Mach regime
University of Ferrara（费拉拉大学）；CNRS and Institut de Mathématiques de Bordeaux (IMB) Université de Bordeaux（法国国家科学研究中心与波尔多大学数学研究所）；Institut de Mathématiques de Toulouse (IMT), Université P. Sabatier（图卢兹大学数学研究所）
A class of high-order shock-capturing schemes, P$_n$T$_m$-BVD (Deng et al., J. Comp. Phys., 386:323-349, 2019; Comput. & Fluids, 200:104433, 2020.) schemes, have been devised to solve the Euler equations with substantially reduced numerical dissipation, which enable high-resolution simulations to resolve flow structures of wider range scales. In such simulations with low dissipation, errors of round-off level might grow and contaminate the numerical solutions. A typical example of such problems is the loss of symmetry in the numerical solutions for physical problems of symmetric configurations…
The total generalized variation extends the total variation by incorporating higher-order smoothness. Thus, it can also suffer from similar discretization issues related to isotropy. Inspired by the success of novel discretization schemes of the total variation, there has been recent work to improve the second-order total generalized variation discretization, based on the same design idea. In this work, we propose to extend this to a general discretization scheme based on interpolation filters, for which we prove variational consistency. We then describe how to learn these interpolation filter…
Hermite parametric surface interpolation based on Argyris element
University of Ljubljana（卢布尔雅那大学）；University of Primorska（普里莫斯卡大学）；INdAM, Unità di Ricerca di Firenze c/o DiMaI “U. Dini”, University of Florence（意大利国家数学研究所佛罗伦萨研究单元迪马伊“乌·迪尼”系，佛罗伦萨大学）
This paper proposes a first-order total variation diminishing (TVD) treatment for coarsening and refining of local timestep size in response to dynamic local variations in wave speeds for nonlinear conservation laws. The algorithm is accompanied with a proof of formal correctness showing that given a sufficiently small minimum timestep the algorithm will produce TVD solution for nonlinear scalar conservation laws. A key feature of the algorithm is its formulation as a discrete event simulation, which allows for easy and efficient parallelization using existing software. Numerical results demon…
Optimal estimation for the Fujino-Morley interpolation error constants
Graduate School of Science and Technology, Niigata University（新潟大学科学技术研究科）；Department of Applied Mathematics, National Cheng Kung University（国立成功大学应用数学系）；Department of Mathematics, National Cheng Kung University（国立成功大学数学系）
This paper proposes two novel approaches to perform more suitable sensitivity analyses for discrete topology optimization methods. To properly support them, we introduce a more formal description of the Bi-directional Evolutionary Structural Optimization (BESO) method, in which the sensitivity analysis is based on finite variations of the objective function. The proposed approaches are compared to a naive strategy; to the conventional strategy, referred to as First-Order Continuous Interpolation (FOCI) approach; and to a strategy previously developed by other researchers, referred to as High-O…
