In this investigation, we propose a numerical method based on the fractional‐order generalized Taylor wavelets (FGTW) for option pricing and the fractional Black–Scholes equations. This model studies option pricing when the underlying asset has subdiffusive dynamics. By applying the regularized beta function, we give an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FGTW. An error analysis of the numerical scheme for estimating solutions is performed. Finally, we conduct a variety of numerical experiments for several standard examples from the literature to assess the efficiency of the proposed method.
{"title":"A wavelet collocation method for fractional Black–Scholes equations by subdiffusive model","authors":"Davood Damircheli, Mohsen Razzaghi","doi":"10.1002/num.23103","DOIUrl":"https://doi.org/10.1002/num.23103","url":null,"abstract":"In this investigation, we propose a numerical method based on the fractional‐order generalized Taylor wavelets (FGTW) for option pricing and the fractional Black–Scholes equations. This model studies option pricing when the underlying asset has subdiffusive dynamics. By applying the regularized beta function, we give an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FGTW. An error analysis of the numerical scheme for estimating solutions is performed. Finally, we conduct a variety of numerical experiments for several standard examples from the literature to assess the efficiency of the proposed method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"45 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140827489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms.
{"title":"Error estimates for completely discrete FEM in energy‐type and weaker norms","authors":"Lutz Angermann, Peter Knabner, Andreas Rupp","doi":"10.1002/num.23106","DOIUrl":"https://doi.org/10.1002/num.23106","url":null,"abstract":"The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"3 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140827488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, a generalized log orthogonal functions (GLOFs)‐spectral collocation method to two dimensional weakly singular Volterra integral equations of the second kind is proposed. The mild singularities of the solution at the interval endpoint can be captured by Gauss‐GLOFs quadrature and the shortcoming of the traditional spectral method which cannot well deal with weakly singular Volterra integral equations with limited regular solutions is avoided. A detailed convergence analysis of the numerical solution is carried out. The efficiency of the proposed method is demonstrated by numerical examples.
{"title":"Generalized log orthogonal functions spectral collocation method for two dimensional weakly singular Volterra integral equations of the second kind","authors":"Qiumei Huang, Min Wang","doi":"10.1002/num.23105","DOIUrl":"https://doi.org/10.1002/num.23105","url":null,"abstract":"In this article, a generalized log orthogonal functions (GLOFs)‐spectral collocation method to two dimensional weakly singular Volterra integral equations of the second kind is proposed. The mild singularities of the solution at the interval endpoint can be captured by Gauss‐GLOFs quadrature and the shortcoming of the traditional spectral method which cannot well deal with weakly singular Volterra integral equations with limited regular solutions is avoided. A detailed convergence analysis of the numerical solution is carried out. The efficiency of the proposed method is demonstrated by numerical examples.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"82 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140827597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper focuses on numerical algorithms to value American options under regime switching. The prices of such options satisfy a set of complementary parabolic problems on an unbounded domain. Based on our previous experience, the pricing model could be truncated into a linear complementarity problem (LCP) over a bounded domain. In addition, we transform the resulting LCP into an equivalent variational problem (VP), and discretize the VP by an Euler‐finite element method. Since the variational matrix in the discretized system is P‐matrix, a primal‐dual active set (PDAS) algorithm is proposed to evaluate the option prices efficiently. As a specialty of PDAS, the optimal exercise boundaries in all regimes are obtained without further computation cost. Finally, numerical simulations are carried out to test the performance of our proposed algorithm and compare it to existing methods.
本文主要研究制度转换下美式期权估值的数值算法。此类期权的价格满足一组无界域上的互补抛物线问题。根据我们以往的经验,定价模型可以截断为有界域上的线性互补问题(LCP)。此外,我们还将 LCP 转化为等效的变分问题(VP),并采用欧拉有限元法对 VP 进行离散化处理。由于离散化系统中的变分矩阵是 P 矩阵,因此我们提出了一种原始双主动集(PDAS)算法来有效评估期权价格。作为 PDAS 的特长,无需更多计算成本即可获得所有制度下的最优行使边界。最后,我们进行了数值模拟,以检验我们提出的算法的性能,并将其与现有方法进行比较。
{"title":"Primal‐dual active set algorithm for valuating American options under regime switching","authors":"Haiming Song, Jingbo Xu, Jinda Yang, Yutian Li","doi":"10.1002/num.23104","DOIUrl":"https://doi.org/10.1002/num.23104","url":null,"abstract":"This paper focuses on numerical algorithms to value American options under regime switching. The prices of such options satisfy a set of complementary parabolic problems on an unbounded domain. Based on our previous experience, the pricing model could be truncated into a linear complementarity problem (LCP) over a bounded domain. In addition, we transform the resulting LCP into an equivalent variational problem (VP), and discretize the VP by an Euler‐finite element method. Since the variational matrix in the discretized system is P‐matrix, a primal‐dual active set (PDAS) algorithm is proposed to evaluate the option prices efficiently. As a specialty of PDAS, the optimal exercise boundaries in all regimes are obtained without further computation cost. Finally, numerical simulations are carried out to test the performance of our proposed algorithm and compare it to existing methods.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"2014 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140628089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we propose ‐unconditional stable schemes for solving time‐dependent incompressible Navier–Stokes equations with smooth or nonsmooth initial data, , . The ‐stability analysis is established by leveraging the scalar auxiliary variable (SAV) approach. When dealing with nonsmooth initial data, we utilize a limited number of iteration of the semi‐implicit scheme followed by the SAV scheme. The overall efficiency is greatly enhanced due to the minimal computational cost of the semi‐implicit scheme and the explicit treatment of the nonlinear term within the SAV approach. The proposed schemes investigate two types of scalar auxiliary variables: the energy‐based variable and the exponential‐based variable. Rigorous proofs of the ‐unconditional stability of both schemes have been provided. Notice that both proposed numerical schemes enjoy unconditional long time stability for smooth and nonsmooth initial data when . Numerical experiments have been conducted to demonstrate the theoretical results.
在本文中,我们提出了用于求解具有光滑或非光滑初始数据(Ⅳ)的时变不可压缩纳维-斯托克斯方程的-条件稳定方案。利用标量辅助变量(SAV)方法建立了-稳定性分析。在处理非光滑初始数据时,我们使用了有限次数的半隐式方案迭代,然后再使用 SAV 方案。由于半隐式方案的计算成本极低,而且 SAV 方法中对非线性项进行了明确处理,因此整体效率大大提高。所提出的方案研究了两类标量辅助变量:基于能量的变量和基于指数的变量。两种方案的无条件稳定性都得到了严格证明。我们注意到,当......或......时,对于光滑或非光滑的初始数据,这两种方案都具有无条件的长时间稳定性。为了证明理论结果,我们进行了数值实验。
{"title":"Unconditional H2$$ {H}^2 $$‐stability of the Euler implicit/explicit SAV‐based scheme for the 2D Navier–Stokes equations with smooth or nonsmooth initial data","authors":"Teng‐Yuan Chang, Ming‐Cheng Shiue","doi":"10.1002/num.23099","DOIUrl":"https://doi.org/10.1002/num.23099","url":null,"abstract":"In this article, we propose ‐unconditional stable schemes for solving time‐dependent incompressible Navier–Stokes equations with smooth or nonsmooth initial data, , . The ‐stability analysis is established by leveraging the scalar auxiliary variable (SAV) approach. When dealing with nonsmooth initial data, we utilize a limited number of iteration of the semi‐implicit scheme followed by the SAV scheme. The overall efficiency is greatly enhanced due to the minimal computational cost of the semi‐implicit scheme and the explicit treatment of the nonlinear term within the SAV approach. The proposed schemes investigate two types of scalar auxiliary variables: the energy‐based variable and the exponential‐based variable. Rigorous proofs of the ‐unconditional stability of both schemes have been provided. Notice that both proposed numerical schemes enjoy unconditional long time stability for smooth and nonsmooth initial data when . Numerical experiments have been conducted to demonstrate the theoretical results.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"48 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose an a posteriori error estimate of the weak Galerkin finite element method (WG-FEM) solving the Stokes problems with variable coefficients. Its error estimator, based on the property of Stokes' law conservation, Helmholtz decomposition and bubble functions, yields global upper bound and local lower bound for the approximation error of the WG-FEM. Error analysis is proved to be valid under the mesh assumptions of the WG-FEM and the way can be extended to other FEMs with the property of Stokes' law conservation, for example, discontinuous Galerkin (DG) FEMs. Finally, we verify the performance of error estimator by performing a few numerical examples.
{"title":"A posteriori error estimate of the weak Galerkin finite element method solving the Stokes problems on polytopal meshes","authors":"Shipeng Xu","doi":"10.1002/num.23102","DOIUrl":"https://doi.org/10.1002/num.23102","url":null,"abstract":"In this paper, we propose an a posteriori error estimate of the weak Galerkin finite element method (WG-FEM) solving the Stokes problems with variable coefficients. Its error estimator, based on the property of Stokes' law conservation, Helmholtz decomposition and bubble functions, yields global upper bound and local lower bound for the approximation error of the WG-FEM. Error analysis is proved to be valid under the mesh assumptions of the WG-FEM and the way can be extended to other FEMs with the property of Stokes' law conservation, for example, discontinuous Galerkin (DG) FEMs. Finally, we verify the performance of error estimator by performing a few numerical examples.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"7 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with the a posteriori error analysis for an augmented mixed discontinuous formulation for the stationary Stokes problem. By considering an appropriate auxiliary problem, we derive an a posteriori error estimator. We prove that this estimator is reliable and locally efficient, and consists of just five residual terms. Numerical experiments confirm the theoretical properties of the augmented discontinuous scheme as well as of the estimator. They also show the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution.
{"title":"An a posteriori error analysis for an augmented discontinuous Galerkin method applied to Stokes problem","authors":"Tomás P. Barrios, Rommel Bustinza","doi":"10.1002/num.23100","DOIUrl":"https://doi.org/10.1002/num.23100","url":null,"abstract":"This paper deals with the a posteriori error analysis for an augmented mixed discontinuous formulation for the stationary Stokes problem. By considering an appropriate auxiliary problem, we derive an a posteriori error estimator. We prove that this estimator is reliable and locally efficient, and consists of just five residual terms. Numerical experiments confirm the theoretical properties of the augmented discontinuous scheme as well as of the estimator. They also show the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"74 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article will suggest a new finite element method to find a ‐velocity and a ‐pressure solving incompressible Stokes equations at low cost. The method solves first the decoupled equation for a ‐velocity. Then, using the calculated velocity, a locally calculable ‐pressure will be defined component‐wisely. The resulting ‐pressure is analyzed to have the optimal order of convergence. Since the pressure is calculated by local computation only, the chief time cost of the new method is on solving the decoupled equation for the ‐velocity. Besides, the method overcomes the problem of singular vertices or corners.
{"title":"A locally calculable P3‐pressure in a decoupled method for incompressible Stokes equations","authors":"Chunjae Park","doi":"10.1002/num.23101","DOIUrl":"https://doi.org/10.1002/num.23101","url":null,"abstract":"This article will suggest a new finite element method to find a ‐velocity and a ‐pressure solving incompressible Stokes equations at low cost. The method solves first the decoupled equation for a ‐velocity. Then, using the calculated velocity, a locally calculable ‐pressure will be defined component‐wisely. The resulting ‐pressure is analyzed to have the optimal order of convergence. Since the pressure is calculated by local computation only, the chief time cost of the new method is on solving the decoupled equation for the ‐velocity. Besides, the method overcomes the problem of singular vertices or corners.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"2015 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196276","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article deals with the error analysis of a Galerkin‐mixed finite element methods for the advection–reaction–diffusion Brinkman flow in porous media. Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest‐order Raviart–Thomas element, the lowest‐order Nédélec edge element and piece‐wise constant discontinuous Galerkin element are used for the velocity, vorticity and pressure, respectively. The existing error estimate of this lowest‐order finite element method is only for all variables in spatial direction, which is not optimal for the concentration variable. This paper focuses on a new and optimal error estimate of a linearized backward Euler Galerkin‐mixed FEMs, where the second‐order accuracy for the concentration in spatial directions is established unconditionally. The key to our optimal error analysis is a new negative norm estimate for Nédélec edge element. Moreover, based on the computed numerical concentration, we propose a simple one‐step recovery technique to obtain a new numerical velocity, vorticity and pressure with second‐order accuracy. Numerical experiments are provided to confirm our theoretical analysis.
{"title":"A new optimal error analysis of a mixed finite element method for advection–diffusion–reaction Brinkman flow","authors":"Huadong Gao, Wen Xie","doi":"10.1002/num.23097","DOIUrl":"https://doi.org/10.1002/num.23097","url":null,"abstract":"This article deals with the error analysis of a Galerkin‐mixed finite element methods for the advection–reaction–diffusion Brinkman flow in porous media. Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest‐order Raviart–Thomas element, the lowest‐order Nédélec edge element and piece‐wise constant discontinuous Galerkin element are used for the velocity, vorticity and pressure, respectively. The existing error estimate of this lowest‐order finite element method is only for all variables in spatial direction, which is not optimal for the concentration variable. This paper focuses on a new and optimal error estimate of a linearized backward Euler Galerkin‐mixed FEMs, where the second‐order accuracy for the concentration in spatial directions is established unconditionally. The key to our optimal error analysis is a new negative norm estimate for Nédélec edge element. Moreover, based on the computed numerical concentration, we propose a simple one‐step recovery technique to obtain a new numerical velocity, vorticity and pressure with second‐order accuracy. Numerical experiments are provided to confirm our theoretical analysis.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"118 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, a first‐order linear fully discrete pressure segregation scheme is studied for the time‐dependent incompressible magnetohydrodynamics (MHD) equations in three‐dimensional bounded domain. Based on an incremental pressure projection method, this scheme allows us to decouple the MHD system into two sub‐problems at each time step, one is the velocity‐magnetic field system, the other is the pressure system. Firstly, a coupled linear elliptic system is solved for the velocity and the magnetic field. Next, a Poisson‐Neumann problem is treated for the pressure. We analyze the temporal error and the spatial error, respectively, and derive the temporal‐spatial error estimates of for the velocity and the magnetic field in the discrete space and for the pressure in the discrete space without imposing constraints on the mesh width and the time step size . Finally, some numerical results are presented to confirm the theoretical predictions and demonstrate the efficiency of the method.
{"title":"Unconditional optimal first‐order error estimates of a full pressure segregation scheme for the magnetohydrodynamics equations","authors":"Yun‐Bo Yang, Yao‐Lin Jiang","doi":"10.1002/num.23098","DOIUrl":"https://doi.org/10.1002/num.23098","url":null,"abstract":"In this article, a first‐order linear fully discrete pressure segregation scheme is studied for the time‐dependent incompressible magnetohydrodynamics (MHD) equations in three‐dimensional bounded domain. Based on an incremental pressure projection method, this scheme allows us to decouple the MHD system into two sub‐problems at each time step, one is the velocity‐magnetic field system, the other is the pressure system. Firstly, a coupled linear elliptic system is solved for the velocity and the magnetic field. Next, a Poisson‐Neumann problem is treated for the pressure. We analyze the temporal error and the spatial error, respectively, and derive the temporal‐spatial error estimates of for the velocity and the magnetic field in the discrete space and for the pressure in the discrete space without imposing constraints on the mesh width and the time step size . Finally, some numerical results are presented to confirm the theoretical predictions and demonstrate the efficiency of the method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"30 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}