Journal of Computational and Applied Mathematics最新文献

{"title":"Pricing of timer volatility-barrier options under Heston’s stochastic volatility model","authors":"Mijin Ha ,&nbsp;Donghyun Kim ,&nbsp;Ji-Hun Yoon","doi":"10.1016/j.cam.2024.116310","DOIUrl":"10.1016/j.cam.2024.116310","url":null,"abstract":"<div><div>Timer options are financial instruments that enable investors to exercise their rights on a random maturity date the realized variance reaches the level of variance budget. These options provide a stable investment opportunity for investors under the unpredictable and complex financial markets, such as global financial crisis or COVID-19 pandemic, which can induce the drastic changes of the volatility for the underlying asset. Meanwhile, in the financial markets, investors who invest in standard timer options may face the problems caused by the postponement of the exercising time for too low volatility, compared to standard vanilla options. In this regard, to overcome such disadvantages, we propose timer volatility-barrier options, which are activated and expired when the volatility arrives at a relatively low barrier level, with the original properties of the standard timer options. In this paper, by making use of the method of images, we derive an analytical formulas for the timer volatility-barrier options so that the volatility process can be driven by Heston stochastic volatility model, and verify the pricing accuracy of the timer options by comparing our solutions with those obtained from Monte Carlo simulations. Finally, we conduct numerical studies on the timer volatility-barrier options to examine the pricing sensitivities with respect to the various model parameters, and implement the discussion for pricing formula of double volatility barrier type of the timer options.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116310"},"PeriodicalIF":2.1,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tail moments and tail joint moments for multivariate generalized hyperbolic distribution","authors":"Yang Yang,&nbsp;Guojing Wang,&nbsp;Jing Yao","doi":"10.1016/j.cam.2024.116307","DOIUrl":"10.1016/j.cam.2024.116307","url":null,"abstract":"<div><div>In this paper, we investigate two novel risk measures under the weighted risk aggregation model: Tail Moment (TM) and Tail Joint Moment (TJM). These measures encompass numerous classical risk measures and are capable of quantifying higher-moment risks such as tail skewness and tail kurtosis. Considering the asymmetric and heavy-tailed properties typical of financial and insurance data, we employ the Multivariate Generalized Hyperbolic (MGH) distribution to model risk variables. Within this framework, we derive analytical expressions for the TM and TJM. These results facilitate the precise assessment of portfolio tail risk as well as the tail dependence between risk assets. Furthermore, we present two applications to highlight the benefits and robustness of TM and TJM in risk management and portfolio selection. In the first example, we utilize tail conditional skewness (TCS) and tail conditional kurtosis (TCK) to evaluate the extreme loss risks of assets, which are not typically captured by conventional risk measure such as marginal expected shortfall (MES) and tail variance (TV). In the second example, we concentrate on the dependence of risks in a downside market. Specifically, we use Tail Correlation (TCOR) and Tail Co-skewness (TCOS) to analyze the risk relationships between stocks and the market index during downturns. These risk measures provide crucial insights for portfolio tail risk assessment and hedging against downside market risk.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116307"},"PeriodicalIF":2.1,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-Fickian diffusion enhanced by temperature","authors":"José A. Ferreira ,&nbsp;Mario Grassi ,&nbsp;Elías Gudiño ,&nbsp;Paula de Oliveira","doi":"10.1016/j.cam.2024.116314","DOIUrl":"10.1016/j.cam.2024.116314","url":null,"abstract":"<div><div>In this paper we present a novel mathematical model to describe the permeation of a fluid through a polymeric matrix, loaded with drug molecules, followed by its subsequent desorption. Both phenomena are enhanced by temperature. We deduce energy estimates and stability estimates for the weak solution of the model, showing that this solution of the problem is stable in bounded time intervals. Numerical simulations illustrate how the coupling effects, of viscoelastic properties and thermal external assistance, can have a central role in the design of drug delivery devices.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116314"},"PeriodicalIF":2.1,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142531789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An MP-Newton method for computing nonlinear eigenpairs and its application for solving a semilinear Schrödinger equation","authors":"Xudong Yao","doi":"10.1016/j.cam.2024.116315","DOIUrl":"10.1016/j.cam.2024.116315","url":null,"abstract":"<div><div>ln Yao and Zhou (2008), a minimax method for computing nonlinear eigenpairs by calculating critical points of the Lagrange multiplier function is presented. But, the method is slow and can find limited amount of eigenpairs. In this paper, a new general characterization, orthogonal-max characterization, for critical points of the Lagrange multiplier function is suggested. An MP-Newton method for finding orthogonal-max type critical points is designed through analyzing how the minimax method works. The new method becomes fast and able to calculate more nonlinear eigenpairs. Numerical experiment confirms these two progresses. Also, the MP-Newton method inherits the advantages of the minimax method. A convergence result for the method is established. Finally, an application for solving a semilinear Schrödinger equation is discussed.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116315"},"PeriodicalIF":2.1,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142531792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysis of a Crank–Nicolson fast element-free Galerkin method for the nonlinear complex Ginzburg–Landau equation","authors":"Xiaolin Li ,&nbsp;Xiyong Cui ,&nbsp;Shougui Zhang","doi":"10.1016/j.cam.2024.116323","DOIUrl":"10.1016/j.cam.2024.116323","url":null,"abstract":"<div><div>A fast element-free Galerkin (EFG) method is proposed in this paper for solving the nonlinear complex Ginzburg–Landau equation. A second-order accurate time semi-discrete system is presented by using the Crank–Nicolson scheme for the temporal discretization, and then a meshless fully discrete system is established by using the EFG method for the spatial discretization. In the proposed EFG method, Nitsche’s technique is used to impose the essential boundary conditions in a weak sense, and the reproducing kernel gradient smoothing integration is used to accelerate the calculation. Theoretical errors for the time semi-discrete system and the fully discrete EFG system are analyzed in detail. Optimal error estimates of the fully discrete Crank–Nicolson EFG method are obtained in both <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> norms. Numerical results validate the theoretical results and the effectiveness of the method.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116323"},"PeriodicalIF":2.1,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142433517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal L2 error estimates of mass- and energy- conserved FE schemes for a nonlinear Schrödinger–type system","authors":"Zhuoyue Zhang,&nbsp;Wentao Cai","doi":"10.1016/j.cam.2024.116313","DOIUrl":"10.1016/j.cam.2024.116313","url":null,"abstract":"<div><div>In this paper, we present an implicit Crank–Nicolson finite element (FE) scheme for solving a nonlinear Schrödinger–type system, which includes Schrödinger–Helmholz system and Schrödinger–Poisson system. In our numerical scheme, we employ an implicit Crank–Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116313"},"PeriodicalIF":2.1,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142437849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A new extension of the core inverse","authors":"D. Mosić ,&nbsp;D.E. Ferreyra","doi":"10.1016/j.cam.2024.116305","DOIUrl":"10.1016/j.cam.2024.116305","url":null,"abstract":"<div><div>There are a number of extensions of the core inverse represented by the products of known generalized inverses, like the core-EP inverse, DMP inverse, GC inverse and so on. However, none of them is an inner inverse of the matrix, and especially for an arbitrary nilpotent matrix, such inverses are always null. Our goal is to use a new technic to introduce an extension of the core inverse that preserves the interesting property of being an inner inverse of the matrix which need not necessarily be the null matrix of a nilpotent matrix. We define the extended core inverse for square complex matrices combining the sum and the difference of three known generalized inverses. Various properties and representations of the extended core inverse are developed. The extended dual core inverse is investigated too. We apply the extended core inverse to solve some systems of linear equations and one minimization problem. A significant normal equation related to least-squares solutions can be solved using the extended core inverse.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116305"},"PeriodicalIF":2.1,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142433435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dispersion analysis of SPH for parabolic equations: High-order kernels against tensile instability","authors":"O.P. Stoyanovskaya ,&nbsp;O.A. Burmistrova ,&nbsp;M.S. Arendarenko ,&nbsp;T.V. Markelova","doi":"10.1016/j.cam.2024.116316","DOIUrl":"10.1016/j.cam.2024.116316","url":null,"abstract":"<div><div>The Smoothed Particle Hydrodynamics (SPH) is a meshless particle-based method mainly used to solve dynamical problems for partial differential equations (PDE). By means of dispersion analysis we investigated four classical SPH-discretizations of parabolic PDE differing by the approximation of Laplacian.</div><div>We derived approximate dispersion relations (ADR) for considered SPH-approximations of the Burgers equation. We demonstrated how the analysis of the ADR allows both studying the approximation and stability of numerical scheme and explaining the features of the method that are known from practice, but are counter-intuitive from the theoretical point of view.</div><div>By means of the mathematical analysis of ADR, the phenomenon of conditional approximation of some schemes under consideration is shown. Moreover, we pioneered in obtaining the necessary condition for the stability of the SPH-approximation of parabolic equations in terms of the Fredholm integral operator applied to the function defined by the kernel of the SPH method. Using this condition, we revealed that passing from the classical second-order kernels to high-order kernels for some schemes leads to the appearance of tensile (short-wave) instability. Among the schemes under consideration, we found the one, for which the necessary condition for the stability of short waves is satisfied both for classical and high-order kernels. The fourth order of approximation in space of this scheme is shown theoretically and confirmed in practice.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116316"},"PeriodicalIF":2.1,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Error bounds for Gauss–Lobatto quadrature of analytic functions on an ellipse","authors":"Hiroshi Sugiura ,&nbsp;Takemitsu Hasegawa","doi":"10.1016/j.cam.2024.116326","DOIUrl":"10.1016/j.cam.2024.116326","url":null,"abstract":"&lt;div&gt;&lt;div&gt;For the (&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;)-point Gauss–Jacobi–Lobatto quadrature to integrals with the Jacobi weight function &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; (&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) over the interval &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, we estimate the location where the kernel of the error functional for functions analytic on an ellipse and its interior in the complex plane attains its maximum modulus. As in our previous work on the Gauss–Jacobi rule, when &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, the location is the intersection point of the ellipse with the real axis in the complex plane. When &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; (the Gegenbauer weight), it is the intersection points with the real axis for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; or for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and with the imaginary axis for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; or for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Here, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; (&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;) is a monotonously decreasing function for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;lim&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Some nume","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116326"},"PeriodicalIF":2.1,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A posteriori error estimates and adaptivity for the IMEX BDF2 method for nonlinear parabolic equations","authors":"Shuo Yang,&nbsp;Liutao Tian,&nbsp;Hongjiong Tian","doi":"10.1016/j.cam.2024.116318","DOIUrl":"10.1016/j.cam.2024.116318","url":null,"abstract":"<div><div>In this paper, we establish optimal a posteriori error estimates for time discretizations by the IMEX two-step backward differentiation formula (BDF2) method for nonlinear parabolic equations. An effective tool for such derivation is appropriate second-order reconstructions of the piecewise linear approximate solution. We employ the second-order reconstructions to establish the upper and lower error bounds which depend only on the data of the problem and the discretization parameters. By means of the a posteriori error estimates, we design a time adaptive algorithm of IMEX BDF2 method. Numerical experiments for the Allen–Cahn equation with smooth and non-smooth initial data are performed to verify our theoretical results and demonstrate the efficiency of the time adaptive algorithm. In addition, we use the IMEX BDF2 method to solve the Navier–Stokes equations to test the validity of the a posteriori error estimates.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116318"},"PeriodicalIF":2.1,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}