Sparse-Grid Implementation of Fixed-Point Fast Sweeping WENO Schemes for Eikonal Equations

Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.

Preface

This focused issue of the Communications on Applied Mathematics and Computation is dedicated to the memory of Professor Ching-Shan Chou,who passed away in November 2021.With her passing,our community of applied mathematicians lost not only a brilliant researcher but also a cherished friend and colleague.

Efficient Sparse-Grid Implementation of a Fifth-Order Multi-resolution WENO Scheme for Hyperbolic Equations

High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.

A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions.A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems.Hence,they are easy to be applied to a general hyperbolic system.To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains,inverse Lax-Wendroff(ILW)procedures were developed as a very effective approach in the literature.In this paper,we combine a fifthorder fixed-point fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing regions.Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids.Numerical results show highorder accuracy and good performance of the method.Furthermore,the method is compared with the popular third-order total variation diminishing Runge-Kutta(TVD-RK3)time-marching method for steady-state computations.Numerical examples show that for most of examples,the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

Preface to the Focused Issue on WENO Schemes

The weighted essentially-oscillatory(WENO)schemes are a class of finite volume and finite difference methods for solving convection-dominated problems,mainly hyperbolic conservation laws.The idea comes from the earlier essentially-oscillatory(ENO)schemes,first developed in[1]in finite volume version and in[6]in finite difference version,for solving hyperbolic conservation laws.

PI3K/Akt/mTOR信号通路相关蛋白在结直肠癌中的表达及与临床病理特征和预后的关系

目的探究PI3K/Akt/mTOR信号通路中PTEN、p-Akt和p-mTOR在结直肠癌中的表达及与临床病理特征和预后的关系。方法选取2016年1月—2017年12月中国人民武装警察部队海警总队医院52例结直肠癌活检标本。采用免疫组织化学SP法检测PI3K-Akt-mTOR信号通路相关蛋白在结肠癌标本及癌旁正常组织标本中的表达;Logistic回归模型分析PI3K-Akt-mTOR信号通路相关蛋白表达与临床病理特征的关系;Kaplan-Meier生存曲线分析PI3K-Akt-mTOR信号通路相关蛋白表达与预后的关系。结果结肠癌组织中PTEN阳性率低于癌旁正常组织(P<0.05),而p-Akt、p-mTOR阳性率均高于癌旁正常组织(P<0.05);淋巴结转移患者的p-Akt和p-mTOR阳性率均高于无淋巴结转移患者(P<0.05);肿瘤浸润达到浆膜层患者的p-Akt阳性率高于未达浆膜层患者(P<0.05);肿瘤低分化患者的p-Akt阳性率高于高、中分化患者(P<0.05)。临床分期Ⅲ、Ⅳ期患者的p-Akt和p-mTOR阳性率均高于Ⅰ、Ⅱ期患者(P<0.05)。结直肠癌患者临床分期Ⅲ、Ⅳ期[OR=4.098(95%CI:2.337,7.134)]是p-Akt阳性表达的影响因素(P<0.05)。PTEN阳性患者的4年生存率为59.62%,高于阴性患者的21.15%(P<0.05);p-Akt阴性患者4年生存率为65.38%,高于阳性患者的19.23%(P<0.05);p-mTOR阴性患者4年生存率为61.54%,高于阳性患者的21.15%(P<0.05)。结论PI3K/Akt/mTOR信号通路相关蛋白的表达与结直肠癌的发展、预后密切相关,p-Akt、p-mTOR有望成为新的结直肠癌治疗靶点和临床指标。

激波与旋涡对干扰研究：激波动力学特性及声波产生机理

采用五阶WENO格式，通过数值求解二维非定常Navier-Stokes方程，系统模拟了激波与旋涡对的干扰过程。我们研究了强度为1．05和1．2两个典型的激波同不同强度的旋涡对的干扰过程。根据激波动力学特性，我们把激波与掠过型旋涡对干扰分成四类，而碰撞型干扰分成七种不同的类型。研究表明，激波与旋涡对干扰过程中，声波产生有三种机制，激波与涡对的干扰，涡对之间的耦合以及激波与声波的干扰。

Reductive recovery of manganese from low-grade manganese dioxide ore using toxic nitrocellulose acid wastewater as reductant

The hydrometallurgical strategy of extracting Mn from low-grade Mn ores has attracted attention for the production of electrolytic manganese metal(EMM). In this work, the reductive dissolution of low-grade Mn O2 ores using toxic nitrocellulose acidic wastewater(NAW) as a reductant was investigated for the first time. Under the optimized conditions of an Mn O2 ore dosage of 100 g·L-1, an ore particle size of-200 mesh, concentrated H2 SO4-to-NAW volume ratio of 0.12, reaction temperature of 90°C, stirring speed at 160 r·min-1, and a contact time of 120 min, the reductive leaching efficiency of Mn and the total organic carbon(TOC) removal efficiency of NAW reached 97.4% and 98.5%, respectively. The residual TOC of 31.6 mg·L-1 did not adversely affect the preparation of EMM. The current process offers a feasible route for the concurrent realization of the reductive leaching of Mn and the treatment of toxic wastewater via a simple one-step process.

Rapid Determination of Polar Herbicides in Soil Samples Using Accelerated Ultrasonic Extraction(AUE) in Combination with Dispersion and In-situ Derivatization

An effective novel ultrasonic extraction procedure coupled with gas chromatography and negative chemical ionization mass spectroscopy has been developed for quantitative recovery of polar herbicides in soil. This rapid one-step sample preparation methodology was named accelerated ultrasonic extraction(AUE), and is based on elevated temperatures, increased power and dispersing intimate contact. Simultaneously, in-situ derivatization was achieved by the addition of derivatization reagent, chelating agent and dispersing agent. The extraction efficiency was enhanced by the multiple applied force and elevated ultrasonic temperature. The in-situ derivatization efficiency was enhanced considerably by the use of ultrasonic energy. Dozens of samples can be extracted simultaneously with this method. The sensitivity was improved because of the remarkable reduced background noise achieved using GC-MS in negative chemical ionization(NCI) mode. The amount of reagent and various ultrasonic parameters, such as ultrasonic energy, ultrasonic time and ultrasonic temperature, were optimized. The reproducibility of replicate soil extraction determination of 9 herbicides in different matrix samples and at different concentrations(n=7) was in the range of 4.9-12.6% of the relative standard deviation. The obtained LOD values ranged between 0.02-0.37 μg/kg for all herbicides. Here, we present an improved ultrasonic extraction procedure, which we have termed AUE, can serve as a rigorous high efficiency preparation methodology for polar organic contaminants and can be applied to solid sample pre-treatment extensively.

Third Order WENO Scheme on Three Dimensional Tetrahedral Meshes

We extend the weighted essentially non-oscillatory(WENO)schemes on two dimensional triangular meshes developed in[7]to three dimensions,and construct a third order finite volume WENO scheme on three dimensional tetrahedral meshes.We use the Lax-Friedrichs monotone flux as building blocks,third order reconstructions made from combinations of linear polynomials which are constructed on diversified small stencils of a tetrahedral mesh,and non-linear weights using smoothness indicators based on the derivatives of these linear polynomials.Numerical examples are given to demonstrate stability and accuracy of the scheme.

A conservative numerical method for the fractional nonlinear Schrodinger equation in two dimensions

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference(WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schr?dinger(FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations(ODEs) in matrix formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson(CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor(c IIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ~2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.

A Continuous Finite Element Method with Homotopy VanishingViscosity for Solving the Static Eikonal Equation

We develop a second-order continuousfinite element method for solving the static Eikonal equation.It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system.More specifically,the homotopy method is utilized to decrease the viscosity coefficient gradually,while Newton’s method is applied to compute the solution for each viscosity coefficient.Newton’s method alone converges for just big enough viscosity coefficients on very coarse grids and for simple 1D examples,but the proposed method is much more robust and guarantees the convergence of the nonlinear solver for all viscosity coefficients and for all examples over all grids.Numerical experiments from 1D to 3D are presented to confirm the second-order convergence and the effectiveness of the proposed method on both structured or unstructured meshes.