Pub Date : 2024-11-15DOI: 10.1016/j.apnum.2024.11.008
Patrick Bammer, Lothar Banz, Andreas Schröder
In this paper a higher-order mixed finite element method for elastoplasticity with linear kinematic hardening is analyzed. Thereby, the non-differentiability of the involved plasticity functional is resolved by a Lagrange multiplier leading to a three field formulation. The finite element discretization is conforming in the displacement field and the plastic strain but potentially non-conforming in the Lagrange multiplier as its Frobenius norm is only constrained in a certain set of Gauss quadrature points. A discrete inf-sup condition with constant 1 and the well posedness of the discrete mixed problem are shown. Moreover, convergence and guaranteed convergence rates are proved with respect to the mesh size and the polynomial degree, which are optimal for the lowest order case. Numerical experiments underline the theoretical results.
{"title":"Mixed finite elements of higher-order in elastoplasticity","authors":"Patrick Bammer, Lothar Banz, Andreas Schröder","doi":"10.1016/j.apnum.2024.11.008","DOIUrl":"10.1016/j.apnum.2024.11.008","url":null,"abstract":"<div><div>In this paper a higher-order mixed finite element method for elastoplasticity with linear kinematic hardening is analyzed. Thereby, the non-differentiability of the involved plasticity functional is resolved by a Lagrange multiplier leading to a three field formulation. The finite element discretization is conforming in the displacement field and the plastic strain but potentially non-conforming in the Lagrange multiplier as its Frobenius norm is only constrained in a certain set of Gauss quadrature points. A discrete inf-sup condition with constant 1 and the well posedness of the discrete mixed problem are shown. Moreover, convergence and guaranteed convergence rates are proved with respect to the mesh size and the polynomial degree, which are optimal for the lowest order case. Numerical experiments underline the theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"209 ","pages":"Pages 38-54"},"PeriodicalIF":2.2,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142697464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-15DOI: 10.1016/j.apnum.2024.11.009
Jia Li , Wei Guan , Shengzhu Shi , Boying Wu
In this paper, we study the local discontinuous Galerkin (LDG) method for one-dimensional nonlinear convection-diffusion equation. In the LDG scheme, local Lax-Friedrichs numerical flux is adopted for the convection term, and a modified central flux is proposed and applied to the nonlinear diffusion coefficient. The modified central flux overcomes the shortcomings of the traditional flux, and it is beneficial in proving the stability of the LDG scheme. By virtue of the Gauss-Radau projections and the local linearization technique, the optimal error estimates are also obtained. Numerical experiments are presented to confirm the validity of the theoretical results.
{"title":"A local discontinuous Galerkin methods with local Lax-Friedrichs flux and modified central flux for one dimensional nonlinear convection-diffusion equation","authors":"Jia Li , Wei Guan , Shengzhu Shi , Boying Wu","doi":"10.1016/j.apnum.2024.11.009","DOIUrl":"10.1016/j.apnum.2024.11.009","url":null,"abstract":"<div><div>In this paper, we study the local discontinuous Galerkin (LDG) method for one-dimensional nonlinear convection-diffusion equation. In the LDG scheme, local Lax-Friedrichs numerical flux is adopted for the convection term, and a modified central flux is proposed and applied to the nonlinear diffusion coefficient. The modified central flux overcomes the shortcomings of the traditional flux, and it is beneficial in proving the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> stability of the LDG scheme. By virtue of the Gauss-Radau projections and the local linearization technique, the optimal error estimates are also obtained. Numerical experiments are presented to confirm the validity of the theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"209 ","pages":"Pages 124-139"},"PeriodicalIF":2.2,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142697460","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}
Pub Date : 2024-11-15DOI: 10.1016/j.apnum.2024.11.007
Guodong Ma , Wei Zhang , Jinbao Jian, Zefeng Huang, Jingyi Mo
The derivative-free projection method (DFPM) is an effective and classic approach for solving the system of nonlinear monotone equations with convex constraints, but the global convergence or convergence rate of the DFPM is typically analyzed under the Lipschitz continuity. This observation motivates us to propose an inertial hybrid DFPM-based algorithm, which incorporates a modified conjugate parameter utilizing a hybridized technique, to weaken the convergence assumption. By integrating an improved inertial extrapolation step and the restart procedure into the search direction, the resulting direction satisfies the sufficient descent and trust region properties, which independent of line search choices. Under weaker conditions, we establish the global convergence and Q-linear convergence rate of the proposed algorithm. To the best of our knowledge, this is the first analysis of the Q-linear convergence rate under the condition that the mapping is locally Lipschitz continuous. Finally, by applying the Bayesian hyperparameter optimization technique, a series of numerical experiment results demonstrate that the new algorithm has advantages in solving nonlinear monotone equation systems with convex constraints and handling compressed sensing problems.
{"title":"An inertial hybrid DFPM-based algorithm for constrained nonlinear equations with applications","authors":"Guodong Ma , Wei Zhang , Jinbao Jian, Zefeng Huang, Jingyi Mo","doi":"10.1016/j.apnum.2024.11.007","DOIUrl":"10.1016/j.apnum.2024.11.007","url":null,"abstract":"<div><div>The derivative-free projection method (DFPM) is an effective and classic approach for solving the system of nonlinear monotone equations with convex constraints, but the global convergence or convergence rate of the DFPM is typically analyzed under the Lipschitz continuity. This observation motivates us to propose an inertial hybrid DFPM-based algorithm, which incorporates a modified conjugate parameter utilizing a hybridized technique, to weaken the convergence assumption. By integrating an improved inertial extrapolation step and the restart procedure into the search direction, the resulting direction satisfies the sufficient descent and trust region properties, which independent of line search choices. Under weaker conditions, we establish the global convergence and Q-linear convergence rate of the proposed algorithm. To the best of our knowledge, this is the first analysis of the Q-linear convergence rate under the condition that the mapping is locally Lipschitz continuous. Finally, by applying the Bayesian hyperparameter optimization technique, a series of numerical experiment results demonstrate that the new algorithm has advantages in solving nonlinear monotone equation systems with convex constraints and handling compressed sensing problems.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"209 ","pages":"Pages 100-123"},"PeriodicalIF":2.2,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698107","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}
Pub Date : 2024-11-14DOI: 10.1016/j.apnum.2024.11.006
Abdulkarim Hassan Ibrahim , Suliman Al-Homidan
Recent research has highlighted the significant performance of multi-step inertial extrapolation in a wide range of algorithmic applications. This paper introduces a derivative-free projection method (DFPM) with a double-inertial extrapolation step for solving large-scale systems of nonlinear equations. The proposed method's global convergence is established under the assumption that the underlying mapping is Lipschitz continuous and satisfies a certain generalized monotonicity assumption (e.g., it can be pseudo-monotone). This is the first convergence result for a DFPM with double inertial step to solve nonlinear equations. Numerical experiments are conducted using well-known test problems to show the proposed method's effectiveness and robustness compared to two existing methods in the literature.
{"title":"A derivative-free projection method with double inertial effects for solving nonlinear equations","authors":"Abdulkarim Hassan Ibrahim , Suliman Al-Homidan","doi":"10.1016/j.apnum.2024.11.006","DOIUrl":"10.1016/j.apnum.2024.11.006","url":null,"abstract":"<div><div>Recent research has highlighted the significant performance of multi-step inertial extrapolation in a wide range of algorithmic applications. This paper introduces a derivative-free projection method (DFPM) with a double-inertial extrapolation step for solving large-scale systems of nonlinear equations. The proposed method's global convergence is established under the assumption that the underlying mapping is Lipschitz continuous and satisfies a certain generalized monotonicity assumption (e.g., it can be pseudo-monotone). This is the first convergence result for a DFPM with double inertial step to solve nonlinear equations. Numerical experiments are conducted using well-known test problems to show the proposed method's effectiveness and robustness compared to two existing methods in the literature.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"209 ","pages":"Pages 55-67"},"PeriodicalIF":2.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142697465","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}
Pub Date : 2024-11-14DOI: 10.1016/j.apnum.2024.11.004
Vittoria Bruni , Rosanna Campagna , Domenico Vitulano
Multicomponent signals play a key role in many application fields, such as biology, audio processing, seismology, air traffic control and security. They are well represented in the time-frequency plane where they are mainly characterized by special curves, called ridges, which carry information about the instantaneous frequency (IF) of each signal component. However, ridges identification usually is a difficult task for signals having interfering components and requires the automatic localization of time-frequency interference regions (IRs). This paper presents a study on the use of the frequency parameter of a hyperbolic-polynomial penalized spline (HP-spline) to predict the presence of interference regions. Since HP-splines are suitably designed for signal regression, it is proved that their frequency parameter can capture the change caused by the interaction between signal components in the time-frequency representation. In addition, the same parameter allows us to define a data-driven approach for IR localization, namely HP-spline Signal Interference Detection (HP-SID) method. Numerical experiments show that the proposed HP-SID can identify specific interference regions for different types of multicomponent signals by means of an efficient algorithm that does not require explicit data regression.
多分量信号在生物、音频处理、地震学、空中交通管制和安全等许多应用领域都发挥着重要作用。多分量信号在时频平面上有很好的表现，其主要特征是特殊曲线（称为脊），其中包含每个信号分量的瞬时频率（IF）信息。然而，对于具有干扰成分的信号来说，脊线识别通常是一项艰巨的任务，需要自动定位时频干扰区域（IR）。本文研究了如何利用双曲-多项式惩罚样条曲线（HP-样条曲线）的频率参数来预测干扰区域的存在。由于 HP 样条适合于信号回归，因此证明了其频率参数可以捕捉时频表示中信号成分之间相互作用所引起的变化。此外，同一参数还允许我们定义一种数据驱动的红外定位方法，即 HP 样条信号干扰检测（HP-SID）方法。数值实验表明，所提出的 HP-SID 可以通过无需明确数据回归的高效算法，识别不同类型多分量信号的特定干扰区域。
{"title":"Multicomponent signals interference detection exploiting HP-splines frequency parameter","authors":"Vittoria Bruni , Rosanna Campagna , Domenico Vitulano","doi":"10.1016/j.apnum.2024.11.004","DOIUrl":"10.1016/j.apnum.2024.11.004","url":null,"abstract":"<div><div>Multicomponent signals play a key role in many application fields, such as biology, audio processing, seismology, air traffic control and security. They are well represented in the time-frequency plane where they are mainly characterized by special curves, called ridges, which carry information about the instantaneous frequency (IF) of each signal component. However, ridges identification usually is a difficult task for signals having interfering components and requires the automatic localization of time-frequency interference regions (IRs). This paper presents a study on the use of the frequency parameter of a hyperbolic-polynomial penalized spline (HP-spline) to predict the presence of interference regions. Since HP-splines are suitably designed for signal regression, it is proved that their frequency parameter can capture the change caused by the interaction between signal components in the time-frequency representation. In addition, the same parameter allows us to define a data-driven approach for IR localization, namely HP-spline Signal Interference Detection (HP-SID) method. Numerical experiments show that the proposed HP-SID can identify specific interference regions for different types of multicomponent signals by means of an efficient algorithm that does not require explicit data regression.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"209 ","pages":"Pages 20-37"},"PeriodicalIF":2.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142697463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-14DOI: 10.1016/j.apnum.2024.11.005
Daniel Acosta-Soba , Francisco Guillén-González , J. Rafael Rodríguez-Galván , Jin Wang
In this paper, we present a new computational framework to approximate a Cahn–Hilliard–Navier–Stokes model with variable density and degenerate mobility that preserves the mass of the mixture, the pointwise bounds of the density and the decreasing energy. This numerical scheme is based on a finite element approximation for the Navier–Stokes fluid flow with discontinuous pressure and an upwind discontinuous Galerkin scheme for the Cahn–Hilliard part. Finally, several numerical experiments such as a convergence test and some well-known benchmark problems are conducted.
{"title":"Property-preserving numerical approximation of a Cahn–Hilliard–Navier–Stokes model with variable density and degenerate mobility","authors":"Daniel Acosta-Soba , Francisco Guillén-González , J. Rafael Rodríguez-Galván , Jin Wang","doi":"10.1016/j.apnum.2024.11.005","DOIUrl":"10.1016/j.apnum.2024.11.005","url":null,"abstract":"<div><div>In this paper, we present a new computational framework to approximate a Cahn–Hilliard–Navier–Stokes model with variable density and degenerate mobility that preserves the mass of the mixture, the pointwise bounds of the density and the decreasing energy. This numerical scheme is based on a finite element approximation for the Navier–Stokes fluid flow with discontinuous pressure and an upwind discontinuous Galerkin scheme for the Cahn–Hilliard part. Finally, several numerical experiments such as a convergence test and some well-known benchmark problems are conducted.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"209 ","pages":"Pages 68-83"},"PeriodicalIF":2.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142697466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1016/j.apnum.2024.11.002
Zhihao Ge , Yanan He
In this paper, we propose a new multiphysics finite element method for a quasi-static poroelasticity model. Firstly, to overcome the displacement locking phenomenon and pressure oscillation, we reformulate the original model into a fluid-fluid coupling problem by introducing new variables-the generalized nonlocal Stokes equations and a diffusion equation, which is a completely new model. Then, we design a fully discrete multiphysics finite element method for the reformulated model-linear finite element pairs for the spatial variables and backward Euler method for time discretization. And we prove that the proposed method is stable without any stabilized term and robust for many parameters and it has the optimal convergence order. Finally, we show some numerical tests to verify the theoretical results.
{"title":"A new multiphysics finite element method for a quasi-static poroelasticity model","authors":"Zhihao Ge , Yanan He","doi":"10.1016/j.apnum.2024.11.002","DOIUrl":"10.1016/j.apnum.2024.11.002","url":null,"abstract":"<div><div>In this paper, we propose a new multiphysics finite element method for a quasi-static poroelasticity model. Firstly, to overcome the displacement locking phenomenon and pressure oscillation, we reformulate the original model into a fluid-fluid coupling problem by introducing new variables-the generalized nonlocal Stokes equations and a diffusion equation, which is a completely new model. Then, we design a fully discrete multiphysics finite element method for the reformulated model-linear finite element pairs for the spatial variables <span><math><mo>(</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo>)</mo></math></span> and backward Euler method for time discretization. And we prove that the proposed method is stable without any stabilized term and robust for many parameters and it has the optimal convergence order. Finally, we show some numerical tests to verify the theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"209 ","pages":"Pages 1-19"},"PeriodicalIF":2.2,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653222","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}
Pub Date : 2024-11-13DOI: 10.1016/j.apnum.2024.11.003
Federico Zullo
We consider the Lommel functions for different values of the parameters . We show that if are half integers, then it is possible to describe these functions with an explicit combination of polynomials and trigonometric functions. The polynomials turn out to give Padé approximants for the trigonometric functions. Numerical properties of the zeros of the polynomials are discussed. Also, when μ is an integer, can be written as an integral involving an explicit combination of trigonometric functions. A closed formula for with μ an integer is given.
{"title":"Lommel functions, Padé approximants and hypergeometric functions","authors":"Federico Zullo","doi":"10.1016/j.apnum.2024.11.003","DOIUrl":"10.1016/j.apnum.2024.11.003","url":null,"abstract":"<div><div>We consider the Lommel functions <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>μ</mi><mo>,</mo><mi>ν</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo></math></span> for different values of the parameters <span><math><mo>(</mo><mi>μ</mi><mo>,</mo><mi>ν</mi><mo>)</mo></math></span>. We show that if <span><math><mo>(</mo><mi>μ</mi><mo>,</mo><mi>ν</mi><mo>)</mo></math></span> are half integers, then it is possible to describe these functions with an explicit combination of polynomials and trigonometric functions. The polynomials turn out to give Padé approximants for the trigonometric functions. Numerical properties of the zeros of the polynomials are discussed. Also, when <em>μ</em> is an integer, <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>μ</mi><mo>,</mo><mi>ν</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo></math></span> can be written as an integral involving an explicit combination of trigonometric functions. A closed formula for <span><math><mmultiscripts><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow><none></none><mprescripts></mprescripts><mrow><mn>2</mn></mrow><none></none></mmultiscripts><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>ν</mi><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>ν</mi><mo>;</mo><mi>μ</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>;</mo><mi>sin</mi><mo></mo><msup><mrow><mo>(</mo><mfrac><mrow><mi>θ</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></math></span> with <em>μ</em> an integer is given.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"209 ","pages":"Pages 275-284"},"PeriodicalIF":2.2,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-07DOI: 10.1016/j.apnum.2024.10.011
Linghua Kong , Songpei Ouyang , Rong Gao , Haiyan Liang
Some combined high order compact (CHOC) schemes are proposed for non-self-adjoint and nonlinear Schrödinger equation (NSANLSE). There are first order and second order spatial derivatives , in the NSANLSE. If one uses classical high order compact schemes to approximate and separately, it will widen the bandwidth in practical coding due to matrix multiplication. This will partly counteract the advantages of high order compact. To overcome the deficiency, one solves the spatial derivatives simultaneously by combining them. In other words, it solves and simultaneously in terms of . The idea is applied to discretize NSANLSE in space. Two efficient numerical schemes are proposed for NSANLSE. The stability and convergence of the new schemes are analyzed theoretically. Numerical experiments are reported to verify the new schemes.
{"title":"Combined high order compact schemes for non-self-adjoint nonlinear Schrödinger equations","authors":"Linghua Kong , Songpei Ouyang , Rong Gao , Haiyan Liang","doi":"10.1016/j.apnum.2024.10.011","DOIUrl":"10.1016/j.apnum.2024.10.011","url":null,"abstract":"<div><div>Some combined high order compact (CHOC) schemes are proposed for non-self-adjoint and nonlinear Schrödinger equation (NSANLSE). There are first order and second order spatial derivatives <span><math><mover><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mo>‾</mo></mover></math></span>, <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub></math></span> in the NSANLSE. If one uses classical high order compact schemes to approximate <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub></math></span> and <span><math><mover><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mo>‾</mo></mover></math></span> separately, it will widen the bandwidth in practical coding due to matrix multiplication. This will partly counteract the advantages of high order compact. To overcome the deficiency, one solves the spatial derivatives simultaneously by combining them. In other words, it solves <span><math><msubsup><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow><mrow><mi>j</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> and <span><math><msubsup><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub></mrow><mrow><mi>j</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> simultaneously in terms of <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>. The idea is applied to discretize NSANLSE in space. Two efficient numerical schemes are proposed for NSANLSE. The stability and convergence of the new schemes are analyzed theoretically. Numerical experiments are reported to verify the new schemes.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"209 ","pages":"Pages 242-257"},"PeriodicalIF":2.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146002","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}
Pub Date : 2024-11-06DOI: 10.1016/j.apnum.2024.10.020
Xue-lei Lin , Sean Hon
In this work, we propose an absolute value block α-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Motivated by the absolute value block circulant preconditioner proposed in McDonald et al. (2018) [40], we propose an absolute value version of the block α-circulant preconditioner. Since the original block α-circulant preconditioner is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Our proposed preconditioner is the first Hermitian positive definite variant of the block α-circulant preconditioner for the concerned wave equations, which fills the gap between block α-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for a properly chosen α the MINRES solver with the proposed preconditioner achieves a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance the concerned problem. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved.
{"title":"A block α-circulant based preconditioned MINRES method for wave equations","authors":"Xue-lei Lin , Sean Hon","doi":"10.1016/j.apnum.2024.10.020","DOIUrl":"10.1016/j.apnum.2024.10.020","url":null,"abstract":"<div><div>In this work, we propose an absolute value block <em>α</em>-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Motivated by the absolute value block circulant preconditioner proposed in McDonald et al. (2018) <span><span>[40]</span></span>, we propose an absolute value version of the block <em>α</em>-circulant preconditioner. Since the original block <em>α</em>-circulant preconditioner is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Our proposed preconditioner is the first Hermitian positive definite variant of the block <em>α</em>-circulant preconditioner for the concerned wave equations, which fills the gap between block <em>α</em>-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for a properly chosen <em>α</em> the MINRES solver with the proposed preconditioner achieves a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance the concerned problem. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"209 ","pages":"Pages 258-274"},"PeriodicalIF":2.2,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143146003","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}
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