We construct a structure preserving non-conforming finite element approximation scheme for the bi-harmonic wave maps into spheres equation. It satisfies a discrete energy law and preserves the non-convex sphere constraint of the continuous problem. The discrete sphere constraint is enforced at the mesh-points via a discrete Lagrange multiplier. This approach restricts the spatial approximation to the (non-conforming) linear finite elements. We show that the numerical approximation converges to the weak solution of the continuous problem in spatial dimension $d=1$. The convergence analysis in dimensions $d>1$ is complicated by the lack of a discrete product rule as well as the low regularity of the numerical approximation in the non-conforming setting. Hence, we show convergence of the numerical approximation in higher-dimensions by introducing additional stabilization terms in the numerical approximation. We present numerical experiments to demonstrate the performance of the proposed numerical approximation and to illustrate the regularizing effect of the bi-Laplacian which prevents the formation of singularities.
{"title":"Numerical approximation of bi-harmonic wave maps into spheres","authors":"Ľubomír Baňas, Sebastian Herr","doi":"arxiv-2409.11366","DOIUrl":"https://doi.org/arxiv-2409.11366","url":null,"abstract":"We construct a structure preserving non-conforming finite element\u0000approximation scheme for the bi-harmonic wave maps into spheres equation. It\u0000satisfies a discrete energy law and preserves the non-convex sphere constraint\u0000of the continuous problem. The discrete sphere constraint is enforced at the\u0000mesh-points via a discrete Lagrange multiplier. This approach restricts the\u0000spatial approximation to the (non-conforming) linear finite elements. We show\u0000that the numerical approximation converges to the weak solution of the\u0000continuous problem in spatial dimension $d=1$. The convergence analysis in\u0000dimensions $d>1$ is complicated by the lack of a discrete product rule as well\u0000as the low regularity of the numerical approximation in the non-conforming\u0000setting. Hence, we show convergence of the numerical approximation in\u0000higher-dimensions by introducing additional stabilization terms in the\u0000numerical approximation. We present numerical experiments to demonstrate the\u0000performance of the proposed numerical approximation and to illustrate the\u0000regularizing effect of the bi-Laplacian which prevents the formation of\u0000singularities.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The discontinuous Galerkin (DG) method and the spectral volume (SV) method are two widely-used numerical methodologies for solving hyperbolic conservation laws. In this paper, we demonstrate that under specific subdivision assumptions, the SV method can be represented in a DG form with a different inner product. Building on this insight, we extend the oscillation-eliminating (OE) technique, recently proposed in [M. Peng, Z. Sun, and K. Wu, {it Mathematics of Computation}, https://doi.org/10.1090/mcom/3998], to develop a new fully-discrete OESV method. The OE technique is non-intrusive, efficient, and straightforward to implement, acting as a simple post-processing filter to effectively suppress spurious oscillations. From a DG perspective, we present a comprehensive framework to theoretically analyze the stability and accuracy of both general Runge-Kutta SV (RKSV) schemes and the novel OESV method. For the linear advection equation, we conduct an energy analysis of the fully-discrete RKSV method, identifying an upwind condition crucial for stability. Furthermore, we establish optimal error estimates for the OESV schemes, overcoming nonlinear challenges through error decomposition and treating the OE procedure as additional source terms in the RKSV schemes. Extensive numerical experiments validate our theoretical findings and demonstrate the effectiveness and robustness of the proposed OESV method. This work enhances the theoretical understanding and practical application of SV schemes for hyperbolic conservation laws, making the OESV method a promising approach for high-resolution simulations.
非连续伽勒金（DG）方法和谱体积（SV）方法是求解双曲守恒定律的两种广泛使用的数值方法。在本文中，我们证明了在特定的细分假设下，SV 方法可以通过不同的inner product 以 DG 形式表示。在此基础上，我们扩展了最近在[M. Peng, Z. Sun, and K. S. and M. P.Peng, Z. Sun, and K. Wu, {itMathematics of Computation}, https://doi.org/10.1090/mcom/3998] 中提出的振荡消除（OE）技术，发展出一种新的全离散 OESV 方法。OE 技术非侵入式、高效且易于实现，可作为一种简单的后处理滤波器来有效抑制杂散振荡。从 DG 的角度，我们提出了一个综合框架，从理论上分析了一般 Runge-Kutta SV (RKSV) 方案和新型 OESV 方法的稳定性和准确性。对于线性平流方程，我们对完全离散的 RKSV 方法进行了能量分析，确定了对稳定性至关重要的上风条件。此外，我们还为 OESV 方案建立了最优误差估计，通过误差分解克服了非线性挑战，并将 OEprocedure 视为 RKSV 方案中的附加源项。广泛的数值实验验证了我们的理论发现，并证明了所提出的 OESV 方法的有效性和鲁棒性。这项工作增强了对双曲守恒定律 SV 方案的理论理解和实际应用，使 OESV 方法成为高分辨率模拟的一种有前途的方法。
{"title":"Spectral Volume from a DG perspective: Oscillation Elimination, Stability, and Optimal Error Estimates","authors":"Zhuoyun Li, Kailiang Wu","doi":"arxiv-2409.10871","DOIUrl":"https://doi.org/arxiv-2409.10871","url":null,"abstract":"The discontinuous Galerkin (DG) method and the spectral volume (SV) method\u0000are two widely-used numerical methodologies for solving hyperbolic conservation\u0000laws. In this paper, we demonstrate that under specific subdivision\u0000assumptions, the SV method can be represented in a DG form with a different\u0000inner product. Building on this insight, we extend the oscillation-eliminating\u0000(OE) technique, recently proposed in [M. Peng, Z. Sun, and K. Wu, {it\u0000Mathematics of Computation}, https://doi.org/10.1090/mcom/3998], to develop a\u0000new fully-discrete OESV method. The OE technique is non-intrusive, efficient,\u0000and straightforward to implement, acting as a simple post-processing filter to\u0000effectively suppress spurious oscillations. From a DG perspective, we present a\u0000comprehensive framework to theoretically analyze the stability and accuracy of\u0000both general Runge-Kutta SV (RKSV) schemes and the novel OESV method. For the\u0000linear advection equation, we conduct an energy analysis of the fully-discrete\u0000RKSV method, identifying an upwind condition crucial for stability.\u0000Furthermore, we establish optimal error estimates for the OESV schemes,\u0000overcoming nonlinear challenges through error decomposition and treating the OE\u0000procedure as additional source terms in the RKSV schemes. Extensive numerical\u0000experiments validate our theoretical findings and demonstrate the effectiveness\u0000and robustness of the proposed OESV method. This work enhances the theoretical\u0000understanding and practical application of SV schemes for hyperbolic\u0000conservation laws, making the OESV method a promising approach for\u0000high-resolution simulations.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper aims to present a local discontinuous Galerkin (LDG) method for solving backward stochastic partial differential equations (BSPDEs) with Neumann boundary conditions. We establish the $L^2$-stability and optimal error estimates of the proposed numerical scheme. Two numerical examples are provided to demonstrate the performance of the LDG method, where we incorporate a deep learning algorithm to address the challenge of the curse of dimensionality in backward stochastic differential equations (BSDEs). The results show the effectiveness and accuracy of the LDG method in tackling BSPDEs with Neumann boundary conditions.
{"title":"Local discontinuous Galerkin method for nonlinear BSPDEs of Neumann boundary conditions with deep backward dynamic programming time-marching","authors":"Yixiang Dai, Yunzhang Li, Jing Zhang","doi":"arxiv-2409.11004","DOIUrl":"https://doi.org/arxiv-2409.11004","url":null,"abstract":"This paper aims to present a local discontinuous Galerkin (LDG) method for\u0000solving backward stochastic partial differential equations (BSPDEs) with\u0000Neumann boundary conditions. We establish the $L^2$-stability and optimal error\u0000estimates of the proposed numerical scheme. Two numerical examples are provided\u0000to demonstrate the performance of the LDG method, where we incorporate a deep\u0000learning algorithm to address the challenge of the curse of dimensionality in\u0000backward stochastic differential equations (BSDEs). The results show the\u0000effectiveness and accuracy of the LDG method in tackling BSPDEs with Neumann\u0000boundary conditions.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present a fractional spectral collocation method for solving a class of weakly singular Volterra integro-differential equations (VDIEs) with proportional delays and cordial operators. Assuming the underlying solutions are in a specific function space, we derive error estimates in the $L^2_{omega^{alpha,beta,lambda}}$ and $L^{infty}$-norms. A rigorous proof reveals that the numerical errors decay exponentially with the appropriate selections of parameters $lambda$. Subsequently, numerical experiments are conducted to validate the effectiveness of the method.
{"title":"A Fractional spectral method for weakly singular Volterra integro-differential equations with delays of the third-kind","authors":"Borui Zhao","doi":"arxiv-2409.10861","DOIUrl":"https://doi.org/arxiv-2409.10861","url":null,"abstract":"In this paper, we present a fractional spectral collocation method for\u0000solving a class of weakly singular Volterra integro-differential equations\u0000(VDIEs) with proportional delays and cordial operators. Assuming the underlying\u0000solutions are in a specific function space, we derive error estimates in the\u0000$L^2_{omega^{alpha,beta,lambda}}$ and $L^{infty}$-norms. A rigorous proof\u0000reveals that the numerical errors decay exponentially with the appropriate\u0000selections of parameters $lambda$. Subsequently, numerical experiments are\u0000conducted to validate the effectiveness of the method.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Biot's consolidation model is a classical model for the evolution of deformable porous media saturated by a fluid and has various interdisciplinary applications. While numerical solution methods to solve poroelasticity by typical schemes such as finite differences, finite volumes or finite elements have been intensely studied, lattice Boltzmann methods for poroelasticity have not been developed yet. In this work, we propose a novel semi-implicit coupling of lattice Boltzmann methods to solve Biot's consolidation model in two dimensions. To this end, we use a single-relaxation-time lattice Boltzmann method for reaction-diffusion equations to solve the Darcy flow and combine it with a recent pseudo-time multi-relaxation-time lattice Boltzmann scheme for quasi-static linear elasticity by Boolakee, Geier and De Lorenzis (2023, DOI: 10.1016/j.cma.2022.115756). The numerical results demonstrate that naive coupling schemes lead to instabilities when the poroelastic system is strongly coupled. However, the newly developed centered coupling scheme using fully explicit and semi-implicit contributions is stable and accurate in all considered cases, even for the Biot--Willis coefficient being one. Furthermore, the numerical results for Terzaghi's consolidation problem and a two-dimensional extension thereof highlight that the scheme is even able to capture discontinuous solutions arising from instantaneous loading.
{"title":"A lattice Boltzmann method for Biot's consolidation model of linear poroelasticity","authors":"Stephan B. Lunowa, Barbara Wohlmuth","doi":"arxiv-2409.11382","DOIUrl":"https://doi.org/arxiv-2409.11382","url":null,"abstract":"Biot's consolidation model is a classical model for the evolution of\u0000deformable porous media saturated by a fluid and has various interdisciplinary\u0000applications. While numerical solution methods to solve poroelasticity by\u0000typical schemes such as finite differences, finite volumes or finite elements\u0000have been intensely studied, lattice Boltzmann methods for poroelasticity have\u0000not been developed yet. In this work, we propose a novel semi-implicit coupling\u0000of lattice Boltzmann methods to solve Biot's consolidation model in two\u0000dimensions. To this end, we use a single-relaxation-time lattice Boltzmann\u0000method for reaction-diffusion equations to solve the Darcy flow and combine it\u0000with a recent pseudo-time multi-relaxation-time lattice Boltzmann scheme for\u0000quasi-static linear elasticity by Boolakee, Geier and De Lorenzis (2023, DOI:\u000010.1016/j.cma.2022.115756). The numerical results demonstrate that naive\u0000coupling schemes lead to instabilities when the poroelastic system is strongly\u0000coupled. However, the newly developed centered coupling scheme using fully\u0000explicit and semi-implicit contributions is stable and accurate in all\u0000considered cases, even for the Biot--Willis coefficient being one. Furthermore,\u0000the numerical results for Terzaghi's consolidation problem and a\u0000two-dimensional extension thereof highlight that the scheme is even able to\u0000capture discontinuous solutions arising from instantaneous loading.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Image processing on surfaces has drawn significant interest in recent years, particularly in the context of denoising. Salt-and-pepper noise is a special type of noise which randomly sets a portion of the image pixels to the minimum or maximum intensity while keeping the others unaffected. In this paper, We propose the L$_p$TV models on triangle meshes to recover images corrupted by salt-and-pepper noise on surfaces. We establish a lower bound for data fitting term of the recovered image. Motivated by the lower bound property, we propose the corresponding algorithm based on the proximal linearization method with the support shrinking strategy. The global convergence of the proposed algorithm is demonstrated. Numerical examples are given to show good performance of the algorithm.
{"title":"Nonconvex models for recovering images corrupted by salt-and-pepper noise on surfaces","authors":"Yuan Liu, Peiqi Yu, Chao Zeng","doi":"arxiv-2409.11139","DOIUrl":"https://doi.org/arxiv-2409.11139","url":null,"abstract":"Image processing on surfaces has drawn significant interest in recent years,\u0000particularly in the context of denoising. Salt-and-pepper noise is a special\u0000type of noise which randomly sets a portion of the image pixels to the minimum\u0000or maximum intensity while keeping the others unaffected. In this paper, We\u0000propose the L$_p$TV models on triangle meshes to recover images corrupted by\u0000salt-and-pepper noise on surfaces. We establish a lower bound for data fitting\u0000term of the recovered image. Motivated by the lower bound property, we propose\u0000the corresponding algorithm based on the proximal linearization method with the\u0000support shrinking strategy. The global convergence of the proposed algorithm is\u0000demonstrated. Numerical examples are given to show good performance of the\u0000algorithm.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nonlinear eigenvalue problems (NEPs) present significant challenges due to their inherent complexity and the limitations of traditional linear eigenvalue theory. This paper addresses these challenges by introducing a nonlinear generalization of the Bauer-Fike theorem, which serves as a foundational result in classical eigenvalue theory. This generalization provides a robust theoretical framework for understanding the sensitivity of eigenvalues in NEPs, extending the applicability of the Bauer-Fike theorem beyond linear cases. Building on this theoretical foundation, we propose novel iterative methods designed to efficiently solve NEPs. These methods leverage the generalized theorem to improve convergence rates and accuracy, making them particularly effective for complex NEPs with dense spectra. The adaptive contour integral method, in particular, is highlighted for its ability to identify multiple eigenvalues within a specified region of the complex plane, even in cases where eigenvalues are closely clustered. The efficacy of the proposed methods is demonstrated through a series of numerical experiments, which illustrate their superior performance compared to existing approaches. These results underscore the practical applicability of our methods in various scientific and engineering contexts. In conclusion, this paper represents a significant advancement in the study of NEPs by providing a unified theoretical framework and effective computational tools, thereby bridging the gap between theory and practice in the field of nonlinear eigenvalue problems.
{"title":"A Nonlinear Generalization of the Bauer-Fike Theorem and Novel Iterative Methods for Solving Nonlinear Eigenvalue Problems","authors":"Ronald Katende","doi":"arxiv-2409.11098","DOIUrl":"https://doi.org/arxiv-2409.11098","url":null,"abstract":"Nonlinear eigenvalue problems (NEPs) present significant challenges due to\u0000their inherent complexity and the limitations of traditional linear eigenvalue\u0000theory. This paper addresses these challenges by introducing a nonlinear\u0000generalization of the Bauer-Fike theorem, which serves as a foundational result\u0000in classical eigenvalue theory. This generalization provides a robust\u0000theoretical framework for understanding the sensitivity of eigenvalues in NEPs,\u0000extending the applicability of the Bauer-Fike theorem beyond linear cases.\u0000Building on this theoretical foundation, we propose novel iterative methods\u0000designed to efficiently solve NEPs. These methods leverage the generalized\u0000theorem to improve convergence rates and accuracy, making them particularly\u0000effective for complex NEPs with dense spectra. The adaptive contour integral\u0000method, in particular, is highlighted for its ability to identify multiple\u0000eigenvalues within a specified region of the complex plane, even in cases where\u0000eigenvalues are closely clustered. The efficacy of the proposed methods is\u0000demonstrated through a series of numerical experiments, which illustrate their\u0000superior performance compared to existing approaches. These results underscore\u0000the practical applicability of our methods in various scientific and\u0000engineering contexts. In conclusion, this paper represents a significant\u0000advancement in the study of NEPs by providing a unified theoretical framework\u0000and effective computational tools, thereby bridging the gap between theory and\u0000practice in the field of nonlinear eigenvalue problems.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A recently proposed node-based uniform strain virtual element method (NVEM) is here extended to small strain elastoplastic solids. In the proposed method, the strain is averaged at the nodes from the strain of surrounding linearly-precise virtual elements using a generalization to virtual elements of the node-based uniform strain approach for finite elements. The averaged strain is then used to sample the weak form at the nodes of the mesh leading to a method in which all the field variables, including state and history-dependent variables, are related to the nodes and thus they are tracked only at these locations during the nonlinear computations. Through various elastoplastic benchmark problems, we demonstrate that the NVEM is locking-free while enabling linearly-precise virtual elements to solve elastoplastic solids with accuracy.
{"title":"A node-based uniform strain virtual element method for elastoplastic solids","authors":"Rodrigo Silva-Valenzuela, Alejandro Ortiz-Bernardin, Edoardo Artioli","doi":"arxiv-2409.10808","DOIUrl":"https://doi.org/arxiv-2409.10808","url":null,"abstract":"A recently proposed node-based uniform strain virtual element method (NVEM)\u0000is here extended to small strain elastoplastic solids. In the proposed method,\u0000the strain is averaged at the nodes from the strain of surrounding\u0000linearly-precise virtual elements using a generalization to virtual elements of\u0000the node-based uniform strain approach for finite elements. The averaged strain\u0000is then used to sample the weak form at the nodes of the mesh leading to a\u0000method in which all the field variables, including state and history-dependent\u0000variables, are related to the nodes and thus they are tracked only at these\u0000locations during the nonlinear computations. Through various elastoplastic\u0000benchmark problems, we demonstrate that the NVEM is locking-free while enabling\u0000linearly-precise virtual elements to solve elastoplastic solids with accuracy.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Solving sparse linear systems is a critical challenge in many scientific and engineering fields, particularly when these systems are severely ill-conditioned. This work aims to provide a comprehensive comparison of various solvers designed for such problems, offering valuable insights and guidance for domain scientists and researchers. We develop the tools required to accurately evaluate the performance and correctness of 16 solvers from 11 state-of-the-art numerical libraries, focusing on their effectiveness in handling ill-conditioned matrices. The solvers were tested on linear systems arising from a coupled hydro-mechanical marker-in-cell geophysical simulation. To address the challenge of computing accurate error bounds on the solution, we introduce the Projected Adam method, a novel algorithm to efficiently compute the condition number of a matrix without relying on eigenvalues or singular values. Our benchmark results demonstrate that Intel oneAPI MKL PARDISO, UMFPACK, and MUMPS are the most reliable solvers for the tested scenarios. This work serves as a resource for selecting appropriate solvers, understanding the impact of condition numbers, and improving the robustness of numerical solutions in practical applications.
{"title":"A Comparison of Sparse Solvers for Severely Ill-Conditioned Linear Systems in Geophysical Marker-In-Cell Simulations","authors":"Marcel Ferrari","doi":"arxiv-2409.11515","DOIUrl":"https://doi.org/arxiv-2409.11515","url":null,"abstract":"Solving sparse linear systems is a critical challenge in many scientific and\u0000engineering fields, particularly when these systems are severely\u0000ill-conditioned. This work aims to provide a comprehensive comparison of\u0000various solvers designed for such problems, offering valuable insights and\u0000guidance for domain scientists and researchers. We develop the tools required\u0000to accurately evaluate the performance and correctness of 16 solvers from 11\u0000state-of-the-art numerical libraries, focusing on their effectiveness in\u0000handling ill-conditioned matrices. The solvers were tested on linear systems\u0000arising from a coupled hydro-mechanical marker-in-cell geophysical simulation.\u0000To address the challenge of computing accurate error bounds on the solution, we\u0000introduce the Projected Adam method, a novel algorithm to efficiently compute\u0000the condition number of a matrix without relying on eigenvalues or singular\u0000values. Our benchmark results demonstrate that Intel oneAPI MKL PARDISO,\u0000UMFPACK, and MUMPS are the most reliable solvers for the tested scenarios. This\u0000work serves as a resource for selecting appropriate solvers, understanding the\u0000impact of condition numbers, and improving the robustness of numerical\u0000solutions in practical applications.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
All the existing entropy stable (ES) schemes for relativistic hydrodynamics (RHD) in the literature were restricted to the ideal equation of state (EOS), which however is often a poor approximation for most relativistic flows due to its inconsistency with the relativistic kinetic theory. This paper develops high-order ES finite difference schemes for RHD with general Synge-type EOS, which encompasses a range of special EOSs. We first establish an entropy pair for the RHD equations with general Synge-type EOS in any space dimensions. We rigorously prove that the found entropy function is strictly convex and derive the associated entropy variables, laying the foundation for designing entropy conservative (EC) and ES schemes. Due to relativistic effects, one cannot explicitly express primitive variables, fluxes, and entropy variables in terms of conservative variables. Consequently, this highly complicates the analysis of the entropy structure of the RHD equations, the investigation of entropy convexity, and the construction of EC numerical fluxes. By using a suitable set of parameter variables, we construct novel two-point EC fluxes in a unified form for general Synge-type EOS. We obtain high-order EC schemes through linear combinations of the two-point EC fluxes. Arbitrarily high-order accurate ES schemes are achieved by incorporating dissipation terms into the EC schemes, based on (weighted) essentially non-oscillatory reconstructions. Additionally, we derive the general dissipation matrix for general Synge-type EOS based on the scaled eigenvectors of the RHD system. We also define a suitable average of the dissipation matrix at the cell interfaces to ensure that the resulting ES schemes can resolve stationary contact discontinuities accurately. Several numerical examples are provided to validate the accuracy and effectiveness of our schemes for RHD with four special EOSs.
现有文献中所有相对论流体力学（RHD）的熵稳定（ES）方案都局限于理想状态方程（EOS），但由于其与相对论动力学理论不一致，对于大多数相对论流来说，理想状态方程往往是一个较差的近似值。本文为具有一般 Synge 型 EOS 的 RHD 开发了高阶 ES 有限差分方案，其中包括一系列特殊的 EOS。我们首先建立了任意空间维度下具有一般 Synge 型 EOS 的 RHD 方程的熵对。我们有力地证明了所发现的熵函数是严格凸函数，并推导出了相关的熵变量，为设计熵保守（EC）和 ES 方案奠定了基础。由于相对论效应，我们无法用保守变量来明确表达原始变量、通量和熵变量。因此，这使得 RHD 方程的熵结构分析、熵凸性研究和 EC 数值通量的构建变得非常复杂。通过使用合适的参数变量集，我们以统一的形式构建了适用于一般 Synge 型 EOS 的新型两点 EC 通量。我们通过两点欧共体通量的线性组合获得高阶欧共体方案。通过将耗散项纳入基于（加权）基本非振荡重构的 EC 方案，实现了任意高阶精确 ES 方案。此外，我们还根据 RHD 系统的比例特征向量，推导出了一般 Synge 型 EOS 的一般耗散矩阵。我们还定义了单元界面处耗散矩阵的适当平均值，以确保所得到的 ES 方案能够准确地解决静态接触不连续性问题。我们提供了几个数值示例来验证我们的方案对于具有四种特殊 EOS 的 RHD 的准确性和有效性。
{"title":"High-order Accurate Entropy Stable Schemes for Relativistic Hydrodynamics with General Synge-type Equation of State","authors":"Linfeng Xu, Shengrong Ding, Kailiang Wu","doi":"arxiv-2409.10872","DOIUrl":"https://doi.org/arxiv-2409.10872","url":null,"abstract":"All the existing entropy stable (ES) schemes for relativistic hydrodynamics\u0000(RHD) in the literature were restricted to the ideal equation of state (EOS),\u0000which however is often a poor approximation for most relativistic flows due to\u0000its inconsistency with the relativistic kinetic theory. This paper develops\u0000high-order ES finite difference schemes for RHD with general Synge-type EOS,\u0000which encompasses a range of special EOSs. We first establish an entropy pair\u0000for the RHD equations with general Synge-type EOS in any space dimensions. We\u0000rigorously prove that the found entropy function is strictly convex and derive\u0000the associated entropy variables, laying the foundation for designing entropy\u0000conservative (EC) and ES schemes. Due to relativistic effects, one cannot\u0000explicitly express primitive variables, fluxes, and entropy variables in terms\u0000of conservative variables. Consequently, this highly complicates the analysis\u0000of the entropy structure of the RHD equations, the investigation of entropy\u0000convexity, and the construction of EC numerical fluxes. By using a suitable set\u0000of parameter variables, we construct novel two-point EC fluxes in a unified\u0000form for general Synge-type EOS. We obtain high-order EC schemes through linear\u0000combinations of the two-point EC fluxes. Arbitrarily high-order accurate ES\u0000schemes are achieved by incorporating dissipation terms into the EC schemes,\u0000based on (weighted) essentially non-oscillatory reconstructions. Additionally,\u0000we derive the general dissipation matrix for general Synge-type EOS based on\u0000the scaled eigenvectors of the RHD system. We also define a suitable average of\u0000the dissipation matrix at the cell interfaces to ensure that the resulting ES\u0000schemes can resolve stationary contact discontinuities accurately. Several\u0000numerical examples are provided to validate the accuracy and effectiveness of\u0000our schemes for RHD with four special EOSs.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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