In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.
在本文中，我们将考虑分布式线性-二次方最优控制问题，该问题涉及高佩克莱特数值的平流-扩散 PDE。在这种情况下，无论是稳定情况还是非稳定情况，都会出现计算不稳定性。在优化系统中采用了流线上风 Petrov-Galerkin 技术，以克服这些令人不快的影响。我们将采用先优化后离散的有限元法离散化方法。关于抛物线情况，我们将考虑一个稳定的时空框架，稳定也将发生在涉及时间导数的双线性形式中。然后，我们将在此离散化过程的基础上建立还原阶模型，并分析两种可能的设置：在线阶段是否也需要稳定化。为了建立状态变量、控制变量和邻接变量的还原基，我们将考虑采用分区方法中的适当正交分解算法。这是第一次在这种情况下将还原阶模型应用于稳定抛物线问题。讨论将通过计算实验来支持，在实验中将研究有限元求解和 ROM 求解之间的相对误差以及各自的计算时间。
{"title":"A Streamline Upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations Under Optimal Control","authors":"Fabio Zoccolan, Maria Strazzullo, Gianluigi Rozza","doi":"10.1515/cmam-2023-0171","DOIUrl":"https://doi.org/10.1515/cmam-2023-0171","url":null,"abstract":"In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an <jats:italic>optimize-then-discretize</jats:italic> approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"54 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140798105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O(hk+1)O(h^{k+1}) when polynomials of degree k≥0kgeq 0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.
混合非连续伽勒金（HDG）方法被应用于非线性抛物线积分微分方程。非线性函数被认为是 Lipschitz 连续的，以分析时间上均匀的先验边界。当使用度数 k ≥ 0 kgeq 0 的多项式来逼近解和通量变量时，引入并使用扩展型 Ritz-Volterra 投影和 HDG 投影作为中间投影，以实现 O ( h k + 1 ) O(h^{k+1}) 的最优阶收敛。通过放宽非线性变量的假设，逐元素后处理可实现超收敛。利用时间方向上的后向欧拉法和正交规则对积分项进行离散化，得出了完全离散的方案及其误差估计。最后，在二维域数值实例的帮助下，获得了计算结果，验证了我们的结果。
{"title":"HDG Method for Nonlinear Parabolic Integro-Differential Equations","authors":"Riya Jain, Sangita Yadav","doi":"10.1515/cmam-2023-0060","DOIUrl":"https://doi.org/10.1515/cmam-2023-0060","url":null,"abstract":"The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>O</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msup> <m:mi>h</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0060_ineq_0001.png\" /> <jats:tex-math>O(h^{k+1})</jats:tex-math> </jats:alternatives> </jats:inline-formula> when polynomials of degree <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0060_ineq_0002.png\" /> <jats:tex-math>kgeq 0</jats:tex-math> </jats:alternatives> </jats:inline-formula> are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"44 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140593326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper introduces the contents of the first of two special issues associated with the 9th International Conference on Computational Methods in Applied Mathematics, which took place from August 29 to September 2, 2022 in Vienna. It comments on the topics and highlights of all twelve papers of the special issue.
{"title":"Computational Methods in Applied Mathematics (CMAM 2022 Conference, Part 1)","authors":"Michael Feischl, Dirk Praetorius, Michele Ruggeri","doi":"10.1515/cmam-2024-0030","DOIUrl":"https://doi.org/10.1515/cmam-2024-0030","url":null,"abstract":"This paper introduces the contents of the first of two special issues associated with the 9th International Conference on Computational Methods in Applied Mathematics, which took place from August 29 to September 2, 2022 in Vienna. It comments on the topics and highlights of all twelve papers of the special issue.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"5 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140593320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 are concerned with an operator-splitting scheme for linear fractional and fractional degenerate stochastic conservation laws driven by multiplicative Lévy noise. More specifically, using a variant of the classical Kružkov doubling of variables approach, we show that the approximate solutions generated by the splitting scheme converge to the unique stochastic entropy solution of the underlying problems. Finally, the convergence analysis is illustrated by several numerical examples.
{"title":"Convergence of an Operator Splitting Scheme for Fractional Conservation Laws with Lévy Noise","authors":"Soumya Ranjan Behera, Ananta K. Majee","doi":"10.1515/cmam-2023-0174","DOIUrl":"https://doi.org/10.1515/cmam-2023-0174","url":null,"abstract":"In this paper, we are concerned with an operator-splitting scheme for linear fractional and fractional degenerate stochastic conservation laws driven by multiplicative Lévy noise. More specifically, using a variant of the classical Kružkov doubling of variables approach, we show that the approximate solutions generated by the splitting scheme converge to the unique stochastic entropy solution of the underlying problems. Finally, the convergence analysis is illustrated by several numerical examples.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"100 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140593366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The objective of this study is to analyze a quasistatic frictional contact problem involving the interaction between a thermo-viscoelastic body and a thermally conductive foundation. The constitutive relation in our investigation is constructed using a fractional Kelvin–Voigt model to describe displacement behavior. Additionally, the heat conduction aspect is governed by a time-fractional derivative parameter that is associated with temperature. The contact is modeled using the Signorini condition, which is a version of Coulomb’s law for dry friction. We develop a variational formulation for the problem and establish the existence of its weak solution using a combination of techniques, including the theory of monotone operators, Caputo derivative, Galerkin method, and the Banach fixed point theorem. To demonstrate the effectiveness of our approach, we include several numerical simulations that showcase the performance of the method.
{"title":"Analysis and Numerical Simulation of Time-Fractional Derivative Contact Problem with Friction in Thermo-Viscoelasticity","authors":"Mustapha Bouallala, EL-Hassan Essoufi, Youssef Ouafik","doi":"10.1515/cmam-2023-0192","DOIUrl":"https://doi.org/10.1515/cmam-2023-0192","url":null,"abstract":"The objective of this study is to analyze a quasistatic frictional contact problem involving the interaction between a thermo-viscoelastic body and a thermally conductive foundation. The constitutive relation in our investigation is constructed using a fractional Kelvin–Voigt model to describe displacement behavior. Additionally, the heat conduction aspect is governed by a time-fractional derivative parameter that is associated with temperature. The contact is modeled using the Signorini condition, which is a version of Coulomb’s law for dry friction. We develop a variational formulation for the problem and establish the existence of its weak solution using a combination of techniques, including the theory of monotone operators, Caputo derivative, Galerkin method, and the Banach fixed point theorem. To demonstrate the effectiveness of our approach, we include several numerical simulations that showcase the performance of the method.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"230 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lisa Davis, Monika Neda, Faranak Pahlevani, Jorge Reyes, Jiajia Waters
This report investigates a stabilization method for first order hyperbolic differential equations applied to DNA transcription modeling. It is known that the usual unstabilized finite element method contains spurious oscillations for nonsmooth solutions. To stabilize the finite element method the authors consider adding to the first order hyperbolic differential system a stabilization term in space and time filtering. Numerical analysis of the stabilized finite element algorithms and computations describing a few biological settings are studied herein.
本报告研究了应用于 DNA 转录建模的一阶双曲微分方程的稳定方法。众所周知，通常的非稳定有限元法包含非光滑解的假振荡。为了稳定有限元方法，作者考虑在一阶双曲微分方程中加入空间和时间滤波稳定项。本文研究了稳定有限元算法的数值分析和描述一些生物环境的计算。
{"title":"A Numerical Study of a Stabilized Hyperbolic Equation Inspired by Models for Bio-Polymerization","authors":"Lisa Davis, Monika Neda, Faranak Pahlevani, Jorge Reyes, Jiajia Waters","doi":"10.1515/cmam-2023-0222","DOIUrl":"https://doi.org/10.1515/cmam-2023-0222","url":null,"abstract":"This report investigates a stabilization method for first order hyperbolic differential equations applied to DNA transcription modeling. It is known that the usual unstabilized finite element method contains spurious oscillations for nonsmooth solutions. To stabilize the finite element method the authors consider adding to the first order hyperbolic differential system a stabilization term in space and time filtering. Numerical analysis of the stabilized finite element algorithms and computations describing a few biological settings are studied herein.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"26 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of our theory, we consider the pseudostress-velocity formulation of Navier–Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.
{"title":"Adaptive Multi-level Algorithm for a Class of Nonlinear Problems","authors":"Dongho Kim, Eun-Jae Park, Boyoon Seo","doi":"10.1515/cmam-2023-0088","DOIUrl":"https://doi.org/10.1515/cmam-2023-0088","url":null,"abstract":"In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of our theory, we consider the pseudostress-velocity formulation of Navier–Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"241 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140009430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically, for different polynomial degrees. For discretization, we employ quad-tree grids, which allow for local refinement in phase-space, and we show exemplary that adaptive methods can efficiently approximate discontinuous solutions. We investigate the behavior of hierarchical error estimators and error estimators based on local averaging.
{"title":"A Phase-Space Discontinuous Galerkin Scheme for the Radiative Transfer Equation in Slab Geometry","authors":"Riccardo Bardin, Fleurianne Bertrand, Olena Palii, Matthias Schlottbom","doi":"10.1515/cmam-2023-0090","DOIUrl":"https://doi.org/10.1515/cmam-2023-0090","url":null,"abstract":"We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically, for different polynomial degrees. For discretization, we employ quad-tree grids, which allow for local refinement in phase-space, and we show exemplary that adaptive methods can efficiently approximate discontinuous solutions. We investigate the behavior of hierarchical error estimators and error estimators based on local averaging.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"73 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139922783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article discusses the quasi-optimality of adaptive nonconforming finite element methods for distributed optimal control problems governed by 𝑚-harmonic operators for m=1,2m=1,2. A variational discretization approach is employed and the state and adjoint variables are discretized using nonconforming finite elements. Error equivalence results at the continuous and discrete levels lead to a priori and a posteriori error estimates for the optimal control problem. The general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality establishes the quasi-optimality. Numerical results demonstrate the theoretically predicted orders of convergence and the efficiency of the adaptive estimator.
本文讨论了在 m = 1 , 2 m=1,2 条件下，由 𝑚-谐波算子控制的分布式最优控制问题的自适应非符合有限元方法的准最优性。采用了变分离散化方法，并使用非符合有限元对状态变量和邻接变量进行离散化。连续和离散层面的误差等价结果导致了最优控制问题的先验和后验误差估计。包括稳定性、还原性、离散可靠性和准正交性在内的一般公理框架确立了准最优性。数值结果证明了理论预测的收敛阶数和自适应估计器的效率。
{"title":"Convergence of Adaptive Crouzeix–Raviart and Morley FEM for Distributed Optimal Control Problems","authors":"Asha K. Dond, Neela Nataraj, Subham Nayak","doi":"10.1515/cmam-2023-0083","DOIUrl":"https://doi.org/10.1515/cmam-2023-0083","url":null,"abstract":"This article discusses the quasi-optimality of adaptive nonconforming finite element methods for distributed optimal control problems governed by 𝑚-harmonic operators for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>m</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0083_ineq_0001.png\" /> <jats:tex-math>m=1,2</jats:tex-math> </jats:alternatives> </jats:inline-formula>. A variational discretization approach is employed and the state and adjoint variables are discretized using nonconforming finite elements. Error equivalence results at the continuous and discrete levels lead to a priori and a posteriori error estimates for the optimal control problem. The general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality establishes the quasi-optimality. Numerical results demonstrate the theoretically predicted orders of convergence and the efficiency of the adaptive estimator.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"45 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139769478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.
{"title":"Weak Convergence of the Rosenbrock Semi-implicit Method for Semilinear Parabolic SPDEs Driven by Additive Noise","authors":"Jean Daniel Mukam, Antoine Tambue","doi":"10.1515/cmam-2023-0055","DOIUrl":"https://doi.org/10.1515/cmam-2023-0055","url":null,"abstract":"This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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