Journal of Scientific Computing

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Adaptive mesh refinement (AMR) and wavelet-based multi-resolution technique, which refine the spatial resolution in regions of interest and coarsen the mesh in other regions, are widely used in scientific computing for higher computational efficiency. The key component of AMR and the multi-resolution method is the adaptation strategy including the regularity estimation. In this paper, a new adaptation strategy for AMR and the multi-resolution method is proposed. Different from AMR, which measures the function regularity based on an empirical gradient operator, and the multi-resolution method with complicated wavelet analysis, the new approach examines the solution smoothness based on the high-order TENO reconstruction [Fu et al., JCP 305(2016): 333-359]. The core idea is that the TENO scheme not only provides the reconstructed data at the cell interface for flux evaluation but also classifies the local flow scales as smooth or nonsmooth on the discretized mesh. Since the scale-separation procedure is achieved in the spectral and characteristic space, the new adaptation strategy is weakly problem-dependent. Moreover, the extra complexity due to the empirical gradient computing or the wavelet analysis is eliminated. Based on Harten's finite-volume multi-resolution method [JCP 115(1994), 319-338], a set of benchmark cases is simulated to validate the performance of the proposed method.
Face IJK shared by the tetrahedrons L^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\user2{L}}$$\end{document} and R^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\user2{R}}$$\end{document}
The MPFA-DNL algorithm
The mesh convergence graphs for the test 4.1–Heterogeneous Diagonal-Anisotropic Media: on the scalar variable (left) and on its gradient (right)
Scalar variable field on the mesh with 17,544 tetrahedra, at y=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{y}} = 0.5$$\end{document}, for test 4.2–Anisotropic Hollow Domain. a Solution with the linear MPFA-D method. b Solution with the new MPFA-DNL. The black regions within the domain indicate overshoots for the scalar variable and the white regions indicate undershoots for the scalar variable
Scalar variable field on the mesh with 15,376 tetrahedra, at y=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y = 0.5$$\end{document}, for test 4.3—Two-Halves Anisotropic Hollow Domain. a Solution with the linear MPFA-D method. b Solution with the new MPFA-DNL. The black regions within the domain indicate overshoots for the scalar variable and the white regions indicate undershoots for the scalar variable
In the present paper, we solve the steady state diffusion equation in 3D domains by means of a cell-centered finite volume method that uses a Multipoint Flux Approximation with a Diamond Stencil and a Non-Linear defect correction strategy (MPFA-DNL) to guarantee the Discrete Maximum Principle (DMP). Our formulation is based in the fact that the flux of MPFA methods can be split into two different parts: a Two Point Flux Approximation (TPFA) component and the Cross-Diffusion Terms (CDT). In the linear MPFA-D method, this split is particularly simple since it lies at the core of the original method construction. In this context, we introduce a non-linear defect correction, aiming to mitigate, whenever necessary, the contributions from the CDT, avoiding, this way, spurious oscillations and DMP violations. Our new MPFA-DNL scheme is locally conservative and capable of dealing with arbitrary anisotropic diffusion tensors and unstructured meshes, without harming the second order convergence rates of the original MPFA-D. To appraise the accuracy and robustness of our formulation, we solve some benchmark problems found in literature. In this paper, we restrict ourselves to tetrahedral meshes, even though, in principle, there is no restriction to extend the method to other polyhedral control volumes. Keywords 3D diffusion problems · Heterogeneous and anisotropic media · Unstructured tetrahedral meshes · Discrete maximum principle (DMP) · MPFA-DNL B T. M. Cavalcante
A new scheme for communication between overset grids using subcells and weighted essentially non oscillatory (WENO) reconstruction for two-dimensional problems has been proposed. The effectiveness of this procedure is demonstrated using the discontinuous Galerkin method. This scheme uses a standard WENO reconstruction by dividing the immediate neighbors into subcells to find the degrees of freedom in cells near the overset interface. This also has the added advantage that it also works as a limiter if a discontinuity passes through the overset interface. Accuracy tests to demonstrate the maintenance of higher order are provided. Results containing shocks are also provided to demonstrate the limiter aspect of the data communication procedure.
Convergence of drag error for the steady isentropic vortex problem governed by the Euler equations. The caption of each sub plot indicates the operator, degree of mapping, and approach for the metrics, respectively. Lagrange mappings are used
Convergence of pressure and drag error for the steady isentropic vortex problem governed by the Euler equations. The caption of each sub plot indicates the operator and type of mapping, respectively. Degree p+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p+1$$\end{document} mappings with the modified approach for the metrics are used
Convergence of lift error for the subsonic channel flow over a Gaussian bump problem governed by the Euler equations. The caption of each sub plot indicates the operator, degree of mapping, and approach for the metrics, respectively. Lagrange mappings are used
Convergence of lift error as a function of grid size based on the total number of grid nodes and core hours for the subsonic channel flow over a Gaussian bump problem governed by the Euler equations. The caption of each sub plot indicates the operator, degree of mapping, and approach for the metrics, respectively. Lagrange mappings are used for the LGL and LG schemes
The goal of this paper is to outline the requirements for obtaining accurate solutions and functionals from high-order tensor-product generalized summation-by-parts discretizations of the steady two-dimensional linear convection and Euler equations on general curved domains. Two procedures for constructing high-order grids using either Lagrange or B-spline mappings are outlined. For the linear convection equation, four discretizations are derived and characterized—two based on the mortar-element approach and two based on the global summation-by-parts-operator approach. It is shown numerically that the schemes are dual consistent, and the requirements for achieving functional superconvergence for each set of methods are outlined. For the Euler equations, a dual-consistent mortar-element discretization is proposed and the practical requirements for obtaining accurate solutions and superconvergent functionals for problems of increasing practical relevance are delineated through theory and numerical examples.
L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document}-errors and convergence rates in time
L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document}- and H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1}$$\end{document}-errors and convergence orders in space
Initial-boundary values for thermal driven cavity flow
The first two columns are the vector fields and streamlines for velocity, and the third and fourth are the contour lines for pressure and temperature, respectively
Energy evolution of γ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (t)$$\end{document}
This paper concerns an efficient finite element method for the natural convection equations with the scalar auxiliary variable approach. The linearly extrapolated Crank-Nicolson techniques are used to discretize nonlinear terms in the Navier–Stokes equations and the heat equation. The induced scalar auxiliary equation is an univariate ordinary equation. With the benefit of the new defined system, the redefined stability of the proposed method is obtained under the fully explicit scheme for the nonlinear terms without the requirement of skew-symmetric trilinear forms. The optimal convergence rates in space for all the variables are proved. The second order convergence rates in time are also obtained. Finally, numerical experiments are provided to support the theoretical analysis and demonstrate the efficiency of the proposed scheme.
Maximum errors of the SOE approximations for the Gaussian kernel with the number of exponentials. Data are shown for five SOE methods: the best rational approximation, the parabolic contour, the hyperbolic contour, the modified Talbot contour and the VPMR. The lines with different color are the linear fitting of the corresponding methods
Maximum errors of the SOE approximations using the VPMR and the Prony of different kernels as functions of P, (a) the Matérn kernel, (b) the power function kernel, (c) the Ewald splitting kernel and (d) the Helmholtz kernel
CPU time as function of t for h=0.05\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h=0.05$$\end{document} (red square), 0.01 (green circle) and 0.005 (blue triangle). The solid lines are linear fitting of these data
We propose an accurate algorithm for a novel sum-of-exponentials (SOE) approximation of kernel functions, and develop a fast algorithm for convolution quadrature based on the SOE, which allows an order N calculation for N time steps of approximating a continuous temporal convolution integral. The SOE method is constructed by a combination of the de la Vallée-Poussin sum for a semi-analytical exponential expansion of a general kernel, and a model reduction technique to minimize the number of exponentials under a given error tolerance. We employ the SOE expansion for the finite part of the splitting convolution kernel so that the convolution integral can be solved as a system of ordinary differential equations. We show that the SOE method works for general kernels with controllable upperbound of positive exponents. Numerical analysis is provided for both the SOE method and the SOE-based convolution quadrature. Numerical results on different kernels demonstrate attractive performance on both accuracy and efficiency of the proposed method.
(Example 1) 1D AC equation. Upper: error of time discretization, (left) Dirichlet boundary condition; (right) Neumann boundary condition. Bottom: discrete energy evolution, (left) Dirichlet boundary condition; (right) Neumann boundary condition
(Example 2) 1D CH equation: left: time accuracy; middle: space accuracy; right: discrete energy evolution
(Example 3) Discrete energy for 2D AC equation with ε2=0.01,τ=1/500\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^2=0.01,\, \tau =1/500$$\end{document}
In this paper, we construct and analyze a fully discrete method for phase-field gradient flows, which uses extrapolated Runge–Kutta with scalar auxiliary variable (RK–SAV) method in time and discontinuous Galerkin (DG) method in space. We propose a novel technique to decouple the system, after which only several elliptic scalar problems with constant coefficients need to be solved independently. Discrete energy decay property of the method is proved for gradient flows. The scheme can be of arbitrarily high order both in time and space, which is demonstrated rigorously for the Allen–Cahn equation and the Cahn–Hilliard equation. More precisely, optimal \(L^2\)-error bound in space and qth-order convergence rate in time are obtained for q-stage extrapolated RK–SAV/DG method. Several numerical experiments are carried out to verify the theoretical results.
Relative l∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{\infty }$$\end{document} error for derivative approximation with iterated and direct multiquadric trigonometric quasi-interpolation method. Here, (DQ)f, (DQ)2f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(DQ)^2f$$\end{document}, (DQ)4f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(DQ)^4f$$\end{document} represent the first order, second order and fourth order derivative approximation respectively with the proposed iterated method, while D2Qf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^2Qf$$\end{document} represents the second order derivative approximation with the direct method [26]
Node set on the torus and approximate solution at time t=0.2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=0.2$$\end{document}
Numerical solutions of Allen–Cahn equation with the initial condition (5.5) at different time levels (ϵ=0.384\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon =0.384$$\end{document}, Δt=1e-5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t=1e-5$$\end{document}). Motion by mean curvature shrinks convex regions asymptotically to a circle
Temporal evolution of numerical solutions of Allen–Cahn equation with the initial condition (5.6) (ϵ=0.384\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon =0.384$$\end{document}, Δt=1e-5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t=1e-5$$\end{document}). The time levels are shown below each figure
Temporal evolution of numerical solutions of Allen–Cahn equation with the initial condition (5.7) (ϵ=0.02\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon =0.02$$\end{document}, Δt=1e-5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta t=1e-5$$\end{document}). The time levels are shown below each figure
The paper first provides an iterated quasi-interpolation scheme for function approximation over periodic domain and then attempts its applications to solve time-dependent surface partial differential equations (PDEs). The key feature of our scheme is that it gives an approximation directly just by taking a weighted average of the available discrete function values evaluated at sampling centers in the periodic domain. As such, it is simple, easy to compute, and the implementation process is stable. Moreover, if the sampling centers distribute uniformly over the periodic domain, it even preserves the same convergence order to all the derivatives. To employ the iterated quasi-interpolation scheme for solving time-dependent surface PDEs, we follow the framework of the method-of-lines. More precisely, we first reformulate the PDEs in terms of the parametric form of the surface. Then we use our quasi-interpolation scheme to approximate both the analytic solution and its spatial derivatives in the reformulated form (of PDEs) to get a semi-discrete ordinary differential equation (ODE) system. Finally, we adopt an appropriate time-integration technique to obtain the full-discrete scheme. As concrete examples, we take the torus for illustration and solve some benchmark reaction-diffusion equations imposed on the torus at the end of the paper. However, the proposed method is general and works on time-dependent PDEs defined on any smooth closed parameterized surfaces without coordinate singularity.
Two harmonic extraction based Jacobi-Davidson (JD) type algorithms are proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair. They are called cross product-free (CPF) and inverse-free (IF) harmonic JDGSVD algorithms, abbreviated as CPF-HJDGSVD and IF-HJDGSVD, respectively. Compared with the standard extraction based JDGSVD algorithm, the harmonic extraction based algorithms converge more regularly and suit better for computing GSVD components corresponding to interior generalized singular values. Thick-restart CPF-HJDGSVD and IF-HJDGSVD algorithms with some deflation and purgation techniques are developed to compute more than one GSVD components. Numerical experiments confirm the superiority of CPF-HJDGSVD and IF-HJDGSVD to the standard extraction based JDGSVD algorithm.
Comparisons in CPU time of fast L1 and direct L1 schemes for solving Example 4.1 with α=1.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1.5$$\end{document}, where M=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M=100$$\end{document}
The numerical solution in maximum-norm of the Algorithm 1 and the Graded-Uniform scheme for Example 4.2 with α=1.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1.5$$\end{document}
Contour plots of the solutions of Algorithm 1 for Example 4.2 at different time with α=1.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1.5$$\end{document}
The variation of time step sizes of the Algorithm 1 and the Graded-Uniform scheme for Example 4.2 with α=1.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =1.5$$\end{document}
We introduce a symmetric fractional-order reduction (SFOR) method to construct numerical algorithms on general nonuniform temporal meshes for semilinear fractional diffusion-wave equations. By using the novel order reduction method, the governing problem is transformed to an equivalent coupled system, where the explicit orders of time-fractional derivatives involved are all α/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha /2$$\end{document}(1<α<2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1<\alpha <2)$$\end{document}. The linearized L1 scheme and Alikhanov scheme are then proposed on general time meshes. Under some reasonable regularity assumptions and weak restrictions on meshes, the optimal convergence is derived for the two kinds of difference schemes by H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2$$\end{document} energy method. An adaptive time stepping strategy which based on the (fast linearized) L1 and Alikhanov algorithms is designed for the semilinear diffusion-wave equations. Numerical examples are provided to confirm the accuracy and efficiency of proposed algorithms.
In this paper, we propose a QSC-L1 method to solve the two-dimensional variable-order time-fractional mobile-immobile diffusion (TF-MID) equations with variably diffusive coefficients, in which the quadratic spline collocation (QSC) method is employed for the spatial discretization, and the classical L1 formula is used for the temporal discretization. We show that the method is unconditionally stable and convergent with first-order in time and second-order in space with respect to some discrete and continuous L 2 norms. Then, combined with the reduced basis technique, an efficient QSC-L1-RB method is proposed to improve the computational efficiency. Numerical examples are attached to verify the convergence orders, and also the method is applied to identify parameters of the variable-order TF-MID equations. Numerical results confirm the contributions of the reduced basis technique, even when the observation data is contaminated by some levels of random noise.
Numerical stability is of critical importance in general circulation models because it affects the design of algorithms, time to solution, and computational costs associated with the simulations, which are very expensive in practice. In this paper we extend the stability analysis for ocean-atmosphere coupling proposed in [Zhang et al., J. Sci. Comput. 84, 44(2020)] to a more realistic model that includes horizontal advection. We analyze various time-stepping strategies. We find that advection has a stabilizing effect in scenarios common to climate models when bulk interface condition and explicit flux coupling are used. We also show that our method can be used to study the stability impact of advection for other interface conditions such as Dirichlet–Neumann conditions.
The schematic diagrams of triangle mesh (left) and polygon mesh (right)
The schematic diagrams of square mesh (left) and deformed square mesh (right)
The schematic diagrams of Th\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_h $$\end{document} with h=1.9028E-01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h=1.9028\hbox {E}-01$$\end{document} (left) and 1.4289E-01 (right) for Example 4.4
The intention of this paper is to study the \(H^1\)-conforming virtual element method for a class of Sobolev equations with variable coefficients. The semi-discrete scheme is constructed on the basis of the symmetric Nitsche’s method to impose the homogeneous and inhomogeneous Dirichlet boundary conditions in a unified way. The fully discrete scheme is obtained with the Crank–Nicolson scheme for temporal approximation. Under some assumptions about the penalty parameter in the Nitsche’s method, the existence and uniqueness of the semi-discrete and fully discrete solutions are analyzed. Furthermore, for the semi-discrete and fully discrete schemes, optimal a priori error estimates in both a mesh size dependent norm and \(L^2\)-norm are proven. Finally, some numerical experiments are carried out to illustrate the theoretical results.
In this paper, we aim to solve the high dimensional stochastic optimal control problem from the view of the stochastic maximum principle via deep learning. By introducing the extended Hamiltonian system which is essentially a Forward Backward Stochastic Differential Equation (FBSDE) with a maximum condition, we reformulate the original control problem as a new one. According to whether the optimal control has an explicit representation, three algorithms are proposed to solve the new control problem. Numerical results for different examples demonstrate the effectiveness of our proposed algorithms, especially in high dimensional cases. And even if the optimal control u~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{u}$$\end{document} in the maximum condition may not be solved explicitly, our algorithms can still deal with the stochastic optimal control problem. An important application of our proposed method is to calculate the sub-linear expectations, which correspond to a kind of fully nonlinear Partial Differential Equations (PDEs).
The subdiffusion equations with a Caputo fractional derivative of order \(\alpha \in (0,1)\) arise in a wide variety of practical problems, which describe the transport processes, in the force-free limit, slower than Brownian diffusion. In this work, we derive the correction schemes of the Lagrange interpolation with degree k (\(k\le 6\)) convolution quadrature, called \(L_k\) approximation, for the subdiffusion. The key step of designing correction algorithm is to calculate the explicit form of the coefficients of \(L_k\) approximation by the polylogarithm function or Bose-Einstein integral. To construct a \(\tau _8\) approximation of Bose-Einstein integral, the desired \((k+1-\alpha )\)th-order convergence rate can be proved for the correction \(L_k\) scheme with nonsmooth data, which is higher than kth-order BDFk method in [Jin, Li, and Zhou, SIAM J. Sci. Comput., 39 (2017), A3129–A3152; Shi and Chen, J. Sci. Comput., (2020) 85:28]. The numerical experiments with spectral method are given to illustrate theoretical results.
In this study, we investigate how low the degree of polynomials can be to construct a stable conservative pair for incompressible Stokes problems that works on general triangulations. We propose a finite element pair that uses a slightly enriched piecewise linear polynomial space for velocity and piecewise constant space for pressure. The pair is illustrated to be a lowest-degree stable conservative pair for Stokes problems on general triangulations.
The accuracy of information transmission while solving domain decomposed problems is crucial to the smooth transition of a solution around the interface/overlapping region. This paper describes a systematical study on an accuracy-enhancing interface treatment algorithm based on the back and forth error compensation and correction method (BFECC). By repetitively employing a low order interpolation technique (usually local 2nd order) 3 times, this algorithm achieves local 3rd order accuracy. Analytical derivation for any dimensions is made, and the “superconvergence” phenomenon (4th order accuracy) is found for specific positioning of the source and target grids. Its limitation has also been studied. With rotated grids or grids of different sizes, this method can still reduce the error but fails to improve the order of the underlying interpolation technique. A series of numerical experiments on meshes with various translations, rotations, spacing ratios, and perturbations have been tested. Different interface treatments applied to cavity flow are compared. The 3D flow example shows that the BFECC method positively impacts the convergence of the domain decomposed problem.
This paper presents a purely numerical factorization of the propagation operator for a generic hyperbolic equation, based on the work of Towne & Colonius in 2015, that does not require heavy analytical development. This method is applied to form one-way equations with the objective of computing the propagation of waves inside a medium. The main advantage of this formulation is that pseudo eigenvectors and eigenvalues matrices are built, leading to the possibility to use the one-way equations into a true amplitude formalism and/or inside a Bremmer series. These two methods allow an extension of the domain of application of the one-way equations when the medium of propagation presents variations along the privileged direction. In particular, these formulations allow to take into account the phenomena of reflection and refraction of the incident wave. Finally numerical results are presented on different 2D situations based on the linearized Euler equations and compared to the results obtained with a full wave resolution. The issues of both the accuracy and the requirements in computational resources of the one-way resolution are also addressed.
At τ=0.015625\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau = 0.015625$$\end{document} and h=0.015625\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h = 0.015625$$\end{document}, the time indicator intensity at T=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = 1$$\end{document} for example 3 obtained using P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}_1$$\end{document} elements
Initial mesh at t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = 0$$\end{document} and corresponding space indicator values
Spatial adapted anisotropic mesh with a constant time step τ=0.125\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau = 0.125$$\end{document} using space estimator at time T=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = 1$$\end{document} and corresponding space indicator values
Initial mesh at t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t = 0$$\end{document} and corresponding time indicator values
Spatial adapted anisotropic mesh with a constant time step τ=0.03125\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau = 0.03125$$\end{document} using time estimator at time T=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = 1$$\end{document} and corresponding time indicator values
Residual-based anisotropic a posteriori error estimates are derived for the parabolic integro-differential equation (PIDE) with smooth kernel in two-dimensions. Based on C 0-conforming piecewise linear elements for spatial discretization, the fully discrete method is achieved after discretizing in time by a two-step backward difference (BDF-2) formula. Reconstruction technique is used to restore the optimal order convergence in temporal direction. Numerical results confirm our theoretical findings.
In this paper, we study nonlinear filtering problems via solving their corresponding Zakai equations. Using the splitting-up technique, we approximate the Zakai equation with two equations consisting of a first-order stochastic partial differential equation and a deterministic second-order partial differential equation. For the splitting-up equations, we use a spectral Galerkin method for the spatial discretization and a finite difference scheme for the temporal discretization. The main results are an error estimate for the semi-discretized scheme with respect to the spatial variable, and an error estimate for the full discretized scheme. To improve the numerical performance, we apply an adaptive technique to accurately locate the support domain of the solution in each time iteration. Finally, we present numerical experiments to demonstrate our theoretical analysis.
This paper studies a generalized version of projected reflected gradient method coupled with inertial extrapolation step to solve variational inequalities in Hilbert spaces. Our proposed method requires one function evaluation and one projection per iteration alongside inertial extrapolation step which is motivated by the desire to devise faster and less computationally expensive iterative methods for variational inequalities. We obtain weak and linear convergence of the sequence of iterates generated by our method under some standard conditions and numerical results are given to show the efficacy of the proposed iterative scheme. Several versions of recently proposed projected reflected gradient methods in the literature are recovered from our method.
We propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.
Spectral accuracy in space for the ETDRK2 scheme
Snapshots of the computed height function u at the indicated times for the parameters Lx=Ly=12.8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_x=L_y=12.8$$\end{document} and ϵ=0.02\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon =0.02$$\end{document}. The hills at early times are not as high as the ones at later times, and similarly with the valley
Evolutions of energy, roughness and width with ϵ=0.02.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon =0.02.$$\end{document}
In this work, we consider the second order exponential time differencing Runge-Kutta (ETDRK) scheme for solving the epitaxial growth model without slope selection. Based on a linear convex splitting of the energy, by using the Fourier collocation approximation for spatial discretization and applying the ETDRK scheme to the split form of the equation, we propose a second order ETDRK (ETDRK2) scheme for the no-slope-selection epitaxial growth model. We prove the preservation of mass conservation of the ETDRK2 scheme and rigorously establish its error estimate. Several numerical experiments are carried out to verify the accuracy of the scheme. We also simulate the coarsening dynamics with small diffusion coefficients to show the theoretical energy decay rate and the growth rates of the surface roughness and the mound width.
An improved neural networks method based on domain decomposition is proposed to solve partial differential equations, which is an extension of the physics informed neural networks (PINNs). Although recent research has shown that PINNs perform effectively in solving partial differential equations, they still have difficulties in solving large-scale complex problems, due to using a single neural network and gradient pathology. In this paper, the proposed approach aims at implementing calculations on sub-domains and improving the expressiveness of neural networks to mitigate gradient pathology. By investigations, it is shown that, although the neural networks structure and the loss function are complicated, the proposed method outperforms the classical PINNs with respect to training effectiveness, computational accuracy, and computational cost.
In Zhu et al. (SIAM J Sci Comput 43: A3009–A3031, 2021), we proposed a new framework of troubled-cell indicator (TCI) using K-means clustering for the discontinuous Galerkin (DG) methods. However, there are two user-tunable parameters in the framework that depend on the polynomial degree of the solution space, the indication variable and even the test problem, which circumscribe the application of the framework. To overcome this drawback, we introduce two simple techniques in this paper: one is to modify the indication variables and the other is to apply a statistical normalization to the modified values. Coupled with four different indication variables, the modified framework is tested via the classical benchmark problems and produces close results under the same setting of the parameters. The discontinuities are overall well captured and the solutions are free of spurious oscillations. The numerical results demonstrate the effectiveness and flexibility of the modified framework and the success in unifying the parameters. Existing TCIs/limiters for the DG methods can be easily implemented into this framework.
‖IP1ub-u0‖0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert I_{P_1}u_b - u_0\Vert _0$$\end{document} for the first eigenfunction
Errors of the first eigenfunction for h=π27\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h = \frac{\pi }{2^7}$$\end{document}
The partition of domain Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}
In this paper, we observe an interesting phenomenon for a weak Galerkin (WG) approximation of eigenvalue problems, that is, we may obtain good approximations for exact eigenvalues from below or above, only through adjusting the global penalty parameter. Based on this observation, a high accuracy algorithm for computing eigenvalues is designed to yield higher convergence order with lower expenses. Some new techniques are developed to analyze upper and lower bound properties of eigenvalues. Numerical results supporting our theory are given.
This note proposes an algorithm for identifying the poles and residues of a meromorphic function from its noisy values on the imaginary axis. The algorithm uses Möbius transform and Prony’s method in the frequency domain. Numerical results are provided to demonstrate the performance of the algorithm.
Discrete-time discrete-state finite Markov chains are versatile mathematical models for a wide range of real-life stochastic processes. One of most common tasks in studies of Markov chains is computation of the stationary distribution. We propose a new general controlled, easily distributed algorithm for this task. The algorithm includes as special cases a wide range of known, very different, and previously disconnected methods including power iterations, versions of Gauss-Southwell formerly restricted to substochastic matrices, and online distributed algorithms. We prove exponential convergence of our method, demonstrate its high efficiency, and derive straightforward control strategies that achieve convergence rates faster than state-of-the-art algorithms.
Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard method can become arbitrarily stiff when characteristic boundary conditions and nonlinear interface conditions are used. Here we propose a new technique that avoids this stiffness by using characteristic variables to “upwind” the boundary and interface treatment. This is done through the introduction of an additional block boundary displacement variable. Using a unified energy, which expresses both the standard as well as characteristic boundary and interface treatment, we show that the resulting scheme has semidiscrete energy stability for the scalar anisotropic wave equation. The theoretical stability results are confirmed with numerical experiments that also demonstrate the accuracy and robustness of the proposed scheme. The numerical results also show that the characteristic scheme has a time step restriction based on standard wave propagation considerations and not the boundary closure.
Based on the auxiliary subspace techniques, a hierarchical basis a posteriori error estimator is proposed for the Stokes problem in two and three dimensions. For the error estimator, we need to solve only two global diagonal linear systems corresponding to the degree of freedom of velocity and pressure respectively, which reduces the computational cost sharply. The upper and lower bounds up to an oscillation term are shown without saturation assumption. Numerical simulations are performed to demonstrate the reliability of the a posteriori error estimator.
We consider a least-squares/relaxation finite element method for the numerical solution of the prescribed Jacobian equation. We look for its solution via a least-squares approach. We introduce a relaxation algorithm that decouples this least-squares problem into a sequence of local nonlinear problems and variational linear problems. We develop dedicated solvers for the algebraic problems based on Newton’s method and we solve the differential problems using mixed low-order finite elements. Various numerical experiments demonstrate the accuracy, efficiency and the robustness of the proposed method, compared for instance to augmented Lagrangian approaches.
This paper considers the general stokes problems applying the Deep learning Galerkin Method (DGM) and gives the convergence of the DGM which contains two parts. First, guided by data and physical laws, depending on the \( L^{2} \) error we construct an objective function and control the performance of the approximation solution by minimizing the objective function in which the prior knowledge of PDEs and data are encoded. Then, we prove the convergence of the neural network to the exact solution. In particular, due to it is mesh free, the DGM can reduce the computational complexity and achieve the competitive results especially in face of the high dimensional problems. With this, compared with traditional numerical methods, numerical results verify the theoretical analysis and show the applicability and effectiveness of the proposed method.
In this paper, the Bogner-Fox-Schmit (BFS) element finite volume methods (FVM) on a suitable Shishkin mesh for the fourth-order singular perturbed elliptic problems are constructed and analyzed . Firstly, under the proposed several equivalent discrete semi-norms, we convert the analysis of stability to the proof of positive definite property for several matrices by element analysis and algebraic techniques. Then we obtain the stability of the BFS element finite volume schemes, which is independent of the aspect ratio of rectangular elements. Secondly, with reasonable assumptions about the structure of the solution, we establish the error estimate of a special interpolation on the Shishkin mesh. Furthermore, based on the stability and interpolation error estimate, we analyze the error estimate of the finite volume methods. The optimal convergence rate for the energy norm N-3+ε12N-2(lnN)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^{-3}+\varepsilon ^{\frac{1}{2}}N^{-2}(\ln N)^2$$\end{document} is obtained by a particular choice of the transition point for the Shishkin mesh. Finally, numerical experiments are presented to confirm the theoretical results.
Color image segmentation is a key technology in image processing. In this paper, a two-stage image segmentation method is proposed that is based on the nonconvex \(L_1/L_2\) approximation of the Mumford-Shah (MS) model. Wherein, the nonconvex regularization term \(L_1/L_2\) on the gradient can approximate the Hausdorff measure and extract more boundary information. The first stage is to solve the nonconvex variant of the MS model and to obtain the smoothed image u. In our framework, we utilize the semi-proximal alternating direction method of multipliers (sPADMM) to sufficiently solve the proposed model. The second stage is segmenting the smoothed u into different phases with thresholds determined by the threshold clustering method. To better deal with the inherent color structures within different channels, we also apply the quaternion representation of the color image. Quantitative and qualitative results demonstrate clearly that our method is better than some state-of-the-art color image segmentation methods.
We consider a time-fractional biharmonic equation involving a Caputo derivative in time of fractional order α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} and a locally Lipschitz continuous nonlinearity. Local and global existence of solutions is discussed and detailed regularity results are provided. A finite element method in space combined with a backward Euler convolution quadrature in time is analyzed. Our objective is to allow initial data of low regularity compared to the number of derivatives occurring in the governing equation. Using a semigroup type approach, error estimates of optimal order are derived for solutions with smooth and nonsmooth initial data. Numerical tests are presented to validate the theoretical results.
In this work, we develop the discontinuous Galerkin method to simulate 1-D cylindrical and spherical compressible multi-medium flows with an immiscible interface. To treat the interface with higher-order accuracy, the modified ghost fluid method is extended to a second-order version with source terms, in which linearly distributed ghost fluid states are constructed. A multi-medium generalized Riemann problem with the geometrical source is constructed to predict the states and the spatial derivatives at the interface. The predicted interface states and spatial derivatives are then employed to define the linearly distributed ghost fluid states. Theoretical analysis shows that the proposed second-order modified ghost fluid method (2nd-MGFM) can effectively eliminate the first order major error term occurring to the interface and accumulating with time when there is interface acceleration. Numerical results exhibit the proposed 2nd-MGFM can suppress overheating at the accelerating wall and pressure dislocation at the accelerating interface very well.
The 4-clustering performance from 20newsgroups dataset
The 3-clustering performance from wine dataset
The 6-clustering performance from dermatology dataset
The 3-clustering performance from iris dataset
The normalized abstract Laplacian tensor of a weighted hypergraph is investigated. The connectivity of the hypergraph is associated with the geometric multiplicity of the smallest eigenvalue of the abstract Laplacian tensor. There is an inequality between the normalized cut of the hypergraph and the second smallest eigenvalue of the abstract Laplacian tensor. An optimization method of the hypergraph clustering is established and analyzed. Numerical examples illustrate that our method is effective.
We analyze a class of meshfree semi-Lagrangian methods for solving advection problems on smooth, closed surfaces with solenoidal velocity field. In particular, we prove the existence of an embedding equation whose corresponding semi-Lagrangian methods yield the ones in the literature for solving problems on surfaces. Our analysis allows us to apply standard bulk domain convergence theories to the surface counterparts. In addition, we provide detailed descriptions for implementing the proposed methods to run on point clouds. After verifying the convergence rates against the theory, we show that the proposed method is a robust building block for more complicated problems, such as advection problems with non-solenoidal velocity field, inviscid Burgers’ equations and systems of reaction advection diffusion equations for pattern formation.
In this paper, we consider the accuracy-enhancement of discontinuous Galerkin (DG) methods for solving partial differential equations (PDEs) with high order spatial derivatives. It is well known that there are highly oscillatory errors for finite element approximations to PDEs that contain hidden superconvergence points. To exploit this information, a Smoothness-Increasing Accuracy-Conserving (SIAC) filter is used to create a superconvergence filtered solution. This is accomplished by convolving the DG approximation against a B-spline kernel. Previous theoretical results about this technique concentrated on first- and second-order equations. However, for linear higher order equations, Yan and Shu (J Sci Comput 17:27–47, 2002) numerically demonstrated that it is possible to improve the accuracy order to \(2k+1\) for local discontinuous Galerkin (LDG) solutions using the SIAC filter. In this work, we firstly provide the theoretical proof for this observation. Furthermore, we prove the accuracy order of the ultra-weak local discontinuous Galerkin (UWLDG) solutions could be improved to \(2k+2-m\) using the SIAC filter, where \(m=[\frac{n}{2}]\), n is the order of PDEs. Finally, we computationally demonstrate that for nonlinear higher order PDEs, we can also obtain a superconvergence approximation with the accuracy order of \(2k+1\) or \(2k+2-m\) by convolving the LDG solution and the UWLDG solution against the SIAC filter, respectively.
Computational complexity test in practice with different dimension n and ground truth two moons data set
Iterative solutions by proposed PGM (Algorithm 1). The abscissa axis and ordinate axis denote the index of each element and the solution value for iterative solutions, respectively
Convergence analysis of Two Moons data set by different schemes
Convergence analysis of MNIST (3 and 5) data set by different schemes
Generalization and sensitivity of step size τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} test using MNIST (digits 3 and 5) data set. X-axis of subfigure (a) is model and τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} for counterpart of subfigure (b). Y-axis represents accuracy
Graph total variation methods have been proved to be powerful tools for unstructured data classification. The existing algorithms, such as MBO (short for Merriman, Bence, and Osher) algorithm, can solve such problems very efficiently with the help of Nyström approximation. However, the strictly theoretical convergence is still unclear due to such approximation. In this paper, we aim at designing a fast operator-splitting algorithm with a low memory footprint and strict convergence guarantee for two-phase unsupervised classification. We first present a general smooth graph total variation model, which mainly consists of four terms, including the Lipschitz-differential regularization term, general double-well potential term, balanced term, and the boundedness constraint. Then the proximal gradient methods without and with acceleration are designed with low computation cost, due to the closed form solution related to proximal operators. The convergence analysis is further investigated under quite mild conditions. We conduct numerical experiments in order to evaluate the performance and convergence of proposed algorithms, on two different data sets including the synthetic two-moons and the MNIST. Namely, the results demonstrate the convergence and robustness of the proposed algorithms.
We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver [Larios-Cárdenas and Gibou, J. Comput. Phys. (May 2022), 10.1016/] that relies on numerical schemes to enable machine-learning operations on demand. In particular, our routine features double predicting to harness curvature symmetry invariance in favor of precision and stability. The core of this solver is a multilayer perceptron trained on circular- and sinusoidal-interface samples. Its role is to quantify the error in numerical curvature approximations and emit corrected estimates for select grid vertices along the free boundary. These corrections arise in response to preprocessed context level-set, curvature, and gradient data. To promote neural capacity, we have adopted sample negative-curvature normalization, reorientation, and reflection-based augmentation. In the same manner, our system incorporates dimensionality reduction, well-balancedness, and regularization to minimize outlying effects. Our training approach is likewise scalable across mesh sizes. For this purpose, we have introduced dimensionless parametrization and probabilistic subsampling during data production. Together, all these elements have improved the accuracy and efficiency of curvature calculations around under-resolved regions. In most experiments, our strategy has outperformed the numerical baseline at twice the number of redistancing steps while requiring only a fraction of the cost.
The domain and a uniform triangulation for the interface problem
Possible configuration for an interface element. a, b, c: Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varGamma }$$\end{document} intersects at an interface at 2, 3, 4 points. The proposed IVE spaces can be defined on almost arbitrary interface element configuration, as discussed in Sect. 3.1. But the construction of IFE spaces and the error analysis will be a little more technical for those general cases. So for simplicity, we will only consider the case in (2a) for the discussion starting from Sect. 3.2
A 1D analog of the comparison used in Lemmas 13 and 16: a and b: u and u~:=uE±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{u}} := u_E^{\pm }$$\end{document} on Kh±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K^{\pm }_h$$\end{document}; c: ΠK±u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varPi }_K^{\pm }u$$\end{document} for H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document} function in Lemma 13; d: ΠK±u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varPi }_K^{\pm } \mathbf{u}$$\end{document} versus ΠK±u~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varPi }_K^{\pm } {\tilde{\mathbf{u}}}$$\end{document} for H(curl)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{H}({\text {curl}})$$\end{document} case, where scalar functions in this figure is illustrated as the lateral view of the tangential component of the vector functions
Errors of the IVE for the H(curl)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{H}({\text {curl}})$$\end{document} interface problem: α+=β+=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ^+ = \beta ^+ = 10$$\end{document} (left) and α+=β+=100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ^+ = \beta ^+ = 100$$\end{document} (right). The dashed lines are the reference lines indicating an optimal convergence of order O(h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(h)$$\end{document}
This article presents an immersed virtual element method for solving a class of interface problems that combines the advantages of both body-fitted mesh methods and unfitted mesh methods. A background body-fitted mesh is generated initially. On those interface elements, virtual element spaces are constructed as solution spaces to local interface problems, and exact sequences can be established for these new spaces involving discontinuous coefficients. The discontinuous coefficients of interface problems are recast as Hodge star operators that are the key to project immersed virtual functions to classic immersed finite element (IFE) functions for computing numerical solutions. An a priori convergence analysis is established robust with respect to the interface location. The proposed method is capable of handling more complicated interface element configuration and provides better performance than the conventional penalty-type IFE method for the \(\mathbf{H}({\text {curl}})\)-interface problem arising from Maxwell equations. It also brings a connection between various methods such as body-fitted methods, IFE methods, virtual element methods, etc.
In this paper we analyze a posteriori error estimates for a mixed formulation of the linear elasticity eigenvalue problem. A posteriori estimators for the nearly and perfectly compressible elasticity spectral problems are proposed. With a post-process argument, we are able to prove reliability and efficiency for the proposed estimators. The numerical method is based in Raviart-Thomas elements to approximate the pseudostress and piecewise polynomials for the displacement. We illustrate our results with numerical tests in two and three dimensions.
In this paper, we propose to combine the fifth-order Hermite weighted essentially non- oscillatory (HWENO) scheme and fast sweeping method (FSM) for the solution of the steady-state SN transport equation in the finite volume framework. It is well-known that the SN transport equation asymptotically converges to a macroscopic diffusion equation in the limit of optically thick systems with small absorption and sources. Numerical methods which can preserve the asymptotic diffusion limit are referred to as asymptotic preserving methods. In the one-dimensional case, we provide the analysis to demonstrate the asymp- totic preserving property of the high order finite volume HWENO method, by showing that its cell-edge and cell-average fluxes possess the thick diffusion limit. A hybrid strategy to compute the nonlinear weights in the HWENO reconstruction is introduced to save compu- tational cost. Extensive one- and two-dimensional numerical experiments are performed to verify the accuracy, asymptotic preserving property and positivity of the proposed HWENO FSM. The proposed HWENO method can also be combined with the Diffusion Synthetic Acceleration algorithm to improve the computational efficiency.
Limited-memory versions of quasi-Newton methods are efficient approaches to solving large-scale optimization problems in a Euclidean space. In particular, a quasi-Newton symmetric rank-one update used in a trust-region setting has proven to be an effective method. In this paper, a limited-memory Riemannian symmetric rank-one trust-region method with a restart strategy is proposed by combining the intrinsic representation of tangent vectors with a recently proposed efficient solver for the Euclidean limited-memory symmetric rank-one trust-region subproblem. The global convergence is established under the assumptions that the vector transport is isometric and the cost function is Lipschitz continuously differentiable. The computational and spatial complexity are analyzed, and detailed implementations are described. The performance of the proposed method is compared to limited-memory Riemannian BFGS method and other state-of-the-art methods using problems from matrix completion, independent component analysis, phase retrieval, and blind deconvolution. The proposed method is novel for problems on Riemannian and Euclidean spaces.
This paper proposes a variant of the Gray Wolf Optimizer (GWO) called the Cross-Dimensional Coordination Gray Wolf Optimizer (CDCGWO), which utilizes a novel learning technique in which all prior best knowledge is gained by candid solutions (wolves) is used to update the best solution (prey positions). This method maintains the wolf's diversity, preventing premature convergence in multimodal optimization tasks. In addition, CDCGWO provides a unique constraint management approach for real-world constrained engineering optimization problems. The CDCGWO's performance on fifteen widely used multimodal numerical test functions, ten complex IEEE CEC06-2019 suit tests, a randomly generated landscape, and twelve constrained real-world optimization problems in a variety of engineering fields, including industrial chemical producer, power system, process design, and synthesis, mechanical design, power-electronic, and livestock feed ration was evaluated. For all 25 numerical functions and 12 engineering problems, the CDCGWO beats all benchmarks and sixteen out of eighteen state-of-the-art algorithms by an average rank of Friedman test of higher than 78 percent, while exceeding jDE100 and DISHchain1e+12 by 21% and 39%, respectively. For all numerical functions and engineering problems, the Bonferroni-Dunn and Holm's tests indicated that CDCGWO is statistically superior to all benchmark and state-of-the-art algorithms, while its performance is statistically equivalent to jDE100 and DISHchain1e+12. The proposed CDCGWO might be utilized to solve challenges involving multimodal search spaces. In addition, compared to rival benchmarks, CDCGWO is suitable for a broader range of engineering applications.
In this paper, we develop an oscillation-free discontinuous Galerkin (OFDG) method for solving the shallow water equations with a non-flat bottom topography. Due to the nonlinear hyperbolic nature of the shallow water equation, the exact solution may contain shock discontinuities; thus, the numerical solution may generate spurious oscillations if no special treatment is taken near the discontinuities. Besides, it is desirable to construct a numerical scheme that possesses the well-balanced property, which preserves exactly the hydrostatic equilibrium solutions up to the machine error. Based on the existing well-balanced DG schemes proposed in Xing (J Comput Phys 257:536–553, 2014), Xing and Shu (Commun Comput Phys 1: 100–134, 2006), we add an extra damping term into these schemes to control the spurious oscillations. One of the advantages of the damping term in suppressing the oscillations is that the damping technique seems to be more convenient for the theoretical analysis, at least in the semi-discrete analysis. With the careful construction of the damping term, the proposed method not only achieves non-oscillatory property without compromising any order of accuracy, but also inherits the the well-balanced property from original schemes. We also present a detailed procedure for constructing the damping term and a theoretical analysis for the preservation of the well-balanced property. Extensive numerical experiments, including one- and two-dimensional space, demonstrate that the proposed method has desired properties without sacrificing any order of accuracy.
We propose, analyze and test a new adaptive penalty scheme that picks the penalty parameter ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} element by element small where ∇·uh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \cdot u^h$$\end{document} is large. We start by analyzing and testing the new scheme on the most simple but interesting setting, the Stokes problem. Finally, we extend and test the algorithm on the incompressible Navier Stokes equation on complex flow problems. Tests indicate that the new adaptive-ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} penalty method algorithm predicts flow behavior accurately. The scheme is developed in the penalty method but also can be used to pick a grad-div stabilization parameter.
We aim to propose a robust and efficient surface reconstruction (SR) scheme for two-dimensional shallow water equations with wet-dry fronts together with adaptive moving mesh methods on irregular quadrangles. The key ingredient of the surface reconstruction is to define Riemann states based on smoothing the water surface or the bottom topography on the cell boundary. The main difficulties in using adaptive moving mesh methods for shallow water equations are to guarantee the positivity of the water depth and the stationary solution near wet-dry fronts. We use a geometrical conservative method to recover the numerical solutions from the mesh of the previous time level and prove positivity-preserving and well-balanced properties. It is a challenging work to preserve stationary solutions for the adaptive moving mesh method when the computational domain contains wet-dry fronts. To overcome this issue, we propose three steps, which consist of redefining the bottom topography on the new meshes, fixing the mesh vertex of the partially flooded cells, and avoiding the extrema of the solutions on the new meshes. The current adaptive SR schemes can maintain the still-water steady state even if the computational domain contains wet-dry fronts and guarantee the water depth to be nonnegative. We illustrate the performance of the current adaptive SR scheme using several classic experiments of two-dimensional shallow water equations with wet-dry fronts.
Degrees of freedom of 2D Pk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_k$$\end{document} Hermite elements, in the case of k=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=3$$\end{document} and k=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=4$$\end{document}
Degrees of freedom of 3D Pk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_k$$\end{document} Argyris elements, in the case of k=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=5$$\end{document} and k=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=6$$\end{document}
Graded grids at different levels for Experiment 6.1, initial grid size h0=1/8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_0 = 1/8$$\end{document}
In the past decade, there are many works on the finite element methods for the fully nonlinear Hamilton–Jacobi–Bellman (HJB) equations with Cordes condition. The linearised systems have large condition numbers, which depend not only on the mesh size but also on the parameters in the Cordes condition. This paper is concerned with the design and analysis of auxiliary space preconditioners for the linearised systems of a C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^0$$\end{document} finite element discretization of HJB equations [Calcolo, 58, 2021]. Based on the stable decomposition on the auxiliary spaces, we propose both the additive and multiplicative preconditioners which converge uniformly in the sense that the resulting condition number is independent of both the number of degrees of freedom and the parameter λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} in Cordes condition. Numerical experiments are carried out to illustrate the efficiency of the proposed preconditioners.
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Wotao Yin
  • University of California, Los Angeles
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Antonio Marquina
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