以作者查詢圖書館館藏

、以作者查詢臺灣博碩士

、以作者查詢全國書目

、勘誤回報

、線上人數：80

、訪客IP：3.141.198.13

姓名	梁長雯(Chang-Wen Liang)  查詢紙本館藏	畢業系所	數學系
論文名稱	(Nonlinear Elimination Preconditioned Space-Time Algorithm for Hyperbolic PDEs: Insights and Explanation Using ODE Theory)
相關論文	★ 非線性塊狀高斯消去牛頓演算法在噴嘴流體的應用 ★ 以平行 Newton-Krylov-Schwarz 演算法解 Poisson-Boltzmann 方程式的有限元素解在膠體科學上的應用 ★ 最小平方有限元素法求解對流擴散方程以及使用Bubble函數的改良 ★ Bifurcation Analysis of Incompressible Sudden Expansion Flows Using Parallel Computing ★ Parallel Jacobi-Davidson Algorithms and Software Developments for Polynomial Eigenvalue Problems in Quantum Dot Simulation ★ An Inexact Newton Method for Drift-DiffusionModel in Semiconductor Device Simulations ★ Numerical Simulation of Three-dimensional Blood Flows in Arteries Using Domain Decomposition Based Scientific Software Packages in Parallel Computers ★ A Parallel Fully Coupled Implicit Domain Decomposition Method for the Stabilized Finite Element Solution of Three-dimensional Unsteady Incompressible Navier-Stokes Equations ★ A Study for Linear Stability Analysis of Incompressible Flows on Parallel Computers ★ Parallel Computation of Acoustic Eigenvalue Problems Using a Polynomial Jacobi-Davidson Method ★ Numerical Study of Algebraic Multigrid Methods for Solving Linear/Nonlinear Elliptic Problems on Sequential and Parallel Computers ★ A Parallel Multilevel Semi-implicit Scheme of Fluid Modeling for Numerical Low-Temperature Plasma Simulation ★ Performance Comparison of Two PETSc-based Eigensolvers for Quadratic PDE Problems ★ A Parallel Two-level Polynomial Jacobi-Davidson Algorithm for Large Sparse Dissipative Acoustic Eigenvalue Problems ★ A Full Space Lagrange-Newton-Krylov Algorithm for Minimum Time Trajectory Optimization ★ Parallel Two-level Patient-specific Numerical Simulation of Three-dimensional Rheological Blood Flows in Branching Arteries
檔案	[Endnote RIS 格式]    [Bibtex 格式]    [相關文章]   [文章引用]   [完整記錄]   [館藏目錄]   至系統瀏覽論文 (2026-7-31以後開放)
摘要(中)	本論文的第一個重點，是開發高效率且穩健對雙曲型偏微分方程求解的平行化演算法，這類問題在計算科學和工程都有廣泛的應用，其解的不連續性和高度局部非線性特性，使得一般針對橢圓偏微分方程設計的迭代法，效能上表現不盡理想，也常有不收斂的問題。傳統上，以使用時間步進方法為基礎的的平行解法，主要是在於每個時間步驟內的平行化，同時在時間步驟之間保持循序進行。然而，這樣的循序步驟可能成為平行化演算法效能的瓶頸。另一方面，隨著現代超級電腦系統運算能力的提升，全耦合時空平行解法越來越受到學界的重視，對於時空演算法，穩健且高效能的大型稀疏非線性方程求解器為演算法成功扮演重要的角色。 不精確牛頓回朔法（Inexact Newton with Backtracking, INB）是大型非線性系統求解常用方法，但在處理我們感興趣的震盪現象發生時，數值解有不連續的情況發生，造成INB收斂速度較慢。為有效解決此問題，我們提出了一種新的非線性消去(Nonlinear Elimination) 預條件法，以提高針對雙曲型偏微分方程解特性的INB的穩健性和效率。透過將非線性消去法作為左或右預條件法，結合INB，包括：自適應非線性消去的INB-ANE（INB-Adaptive Nonlinear Elimination）和非線性消除預條件的不精確牛頓法（ Nonlinear Elimination Preconditioned Inexact Newton, NEPIN）。這些方法旨在減少高度局部非線性的影響，並為下一次牛頓迭代提供更好的初始猜測。數值實驗結果顯示，這一改進增強了INB在Burgers′和Buckley-Leverett方程中的穩健性，而NEPIN優於INB-ANE，在於前者能正確的偵測震波位置並減少子空間修正後的界面污染對方法收斂性的影響。此外，對於NEPIN而言，收斂所需的牛頓迭代次數幾乎與時間步長和網格大小無關。 儘管數值結果顯示，非線性預條件法是改進牛頓法穩健性和效率有效的技術，但對這類方法的理論分析或啟發性理解的完備，仍有待努力。因此，本論文的第二個重點是提出一種新的衡量方法，稱為剛性比，這一想法是源自剛性常微分方程（ODE），用以量化系統中的非線性不平衡的程度。這一度量類似於線性系統中用於評估矩陣質量的條件數。這指標可以用於衡量對非線性方程使用牛頓法求解的難易程度，或作為開發針對特定應用的預條件器，提供了有價值的設計指引或分析方法的工具。我們推導出牛頓法對應的常微分方程系統並計算其剛性比，我們透過數值實例驗證了，NEPIN中的非線性消除去預條件法有效的降低相應牛頓ODE系統剛性比，對方法的成功有較為清楚的理解。
摘要(英)	As the first focus of this work, we aim to develop an efficient and robust solution algorithm for solving hyperbolic partial differential equations exhibiting solution discontinuity and highly local nonlinearity behavior. Such problems arise in numerous applications in computational science and engineering. Traditionally, parallel solution algorithms using time-marching methods focused on parallelizing computations within each time step while maintaining a sequential approach between time steps. Such a sequential step is the potential bottleneck of parallel algorithm performance. On the other hand, with the increasing computational power of modern parallel computer systems, there has been a growing interest in fully coupled space-time solution algorithms for time-dependent partial differential equations (PDEs), enabling parallelism in the temporal domain. To make the solution algorithm practical, it is crucial to have a robust and efficient nonlinear solver for large, sparse systems of equations in space-time algorithms. The inexact Newton with backtracking (INB) method is a well-known approach for solving nonlinear equations, but it exhibits slow convergence in the presence of shocks of our interests. In this work, we propose a new variant of nonlinear elimination preconditioning techniques to enhance the robustness and improve the efficiency of INB based on the solution characteristics of hyperbolic PDEs. By employing nonlinear elimination as a right or left preconditioner in conjunction with INB, we introduce two methods: INB with adaptive nonlinear elimination (INB-ANE) and nonlinear elimination preconditioned inexact Newton (NEPIN). These methods aim to reduce the impact of highly local nonlinearity and provide a better initial guess for the next Newton iteration. Numerical results show that the modification enhances the robustness of INB for Burgers′ and Buckley-Leverett equations and reveal that NEPIN outperforms INB-ANE in identifying the correct shock location and introducing less interface pollution after the subspace correction before the global update. Additionally, the number of Newton iterations required to converge for NEPIN is almost independent of the time step and mesh size. Although nonlinear preconditioning has been shown numerically to be a valuable technique to enhance the robustness and efficiency of Newton-type methods, a comprehensive theoretical or heuristic understanding of this technique still needs to be completed. Therefore, the second focus of this thesis is to introduce a new metric known as the stiffness ratio, which is borrowed from stiff ordinary differential equations (ODEs) to quantify the unbalanced nature of nonlinear systems. This metric is analogous to the condition number of the coefficient matrix for linear systems, which is used to assess matrix quality. Such insights offer valuable guidance and analytical tools for developing preconditioners tailored to specific applications. An investigation of a highly nonlinear algebraic system reveals that NEPIN reduces stiffness compared to INB. To illustrate this concept, we present numerical examples demonstrating the reduction of stiffness ratios of the corresponding Newton ODE system through nonlinear elimination.
關鍵字(中)	★ 非線性預條件 ★ 時空演算法 ★ 雙曲型偏微分方程	關鍵字(英)	★ Nonlinear Precondition ★ Space-Time Algorithm ★ Hyperbolic PDEs
論文目次	Chinese Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i English Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Mathematial models, their discretizaton, and some test problems . . . 6 2.1 First-order scalar hyperbolic PDE models . . . . . . . . . . . . . . . . . . 6 2.2 Discretization and fully coupled nonlinear system . . . . . . . . . . . . . . 10 2.3 Test problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Nonlinear elimination preconditioning technique for hyperbolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 A review of inexact Newton with backtracking method . . . . . . . . . . . 19 3.2 Nonlinear elimination preconditioning technique . . . . . . . . . . . . . . . 22 3.2.1 Traditional INB with adaptive nonlinear elimination (INB-ANE) . 23 3.2.2 Nonlinear elimination preconditioned inexact Newton (NEPIN) . . 25 3.2.3 Challenges of nonlinear elimination in hyperbolic equations . . . . . 30 3.2.4 Remedies: coarse grid correction and morphological closing . . . . . 31 3.2.5 Two layers nonlinear elimination preconditioning technique . . . . . 35 3.3 Numerical results and discussions . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.1 Stopping condition and other parameter setting . . . . . . . . . . . 37 3.3.2 Efficiency comparison of INB-ANE and NEPIN and their variants for Burger’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.3 Elimination component distribution analysis . . . . . . . . . . . . . 40 3.3.4 The performance of one-layer and two-layer nonlinear elimination preconditioned methods for two-phase flow problems . . . . . . . . 42 4 Insight of nonlinear elimination preconditioning from ODE theory viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 The stiff ordinary differential equation . . . . . . . . . . . . . . . . . . . . 47 4.3 Definition of stiffness ratios for ODE systems . . . . . . . . . . . . . . . . 52 4.4 The connection between Newton’s method and ODE theory . . . . . . . . 57 4.5 QZ method for computing the stiffness ratio . . . . . . . . . . . . . . . . . 59 4.6 Numerical examples and discussions . . . . . . . . . . . . . . . . . . . . . . 68 4.6.1 A simple two-variables problem with unbalanced nonlinearity . . . 69 4.6.2 Poisson-Boltzmann problem . . . . . . . . . . . . . . . . . . . . . . 73 4.6.3 Nozzle flow problem . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.6.4 Space-time formulation for Burgers’ and two-phase flow problems . 82 5 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
參考文獻	[1] R. J. LeVeque. Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM, Philadelphia, 2007. [2] D. S. Balsara. An analysis of the hyperbolic nature of the equations of radiation hydrodynamics. Journal of Quantitative Spectroscopy and Radiative Transfer, 61:617–627, 1999. [3] A. Jameson. The evolution of computational methods in aerodynamics. Journal of Applied Mechanics, 50:1052–1070, 1983. [4] G. Boillat. Nonlinear electrodynamics: Lagrangians and equations of motion. Journal of Mathematical Physics, 11:941–951, 1970. [5] R. M. Colombo. Hyperbolic phase transitions in traffic flow. SIAM Journal on Applied Mathematics, 63:708–721, 2003. [6] X. Dai and Y. Maday. Stable parareal in time method for first-and second-order hyperbolic systems. SIAM Journal on Scientific Computing, 35:A52–A78, 2013. [7] M. J. Gander and S. Vandewalle. Analysis of the parareal time-parallel timeintegration method. SIAM Journal on Scientific Computing, 29:556–578, 2007. [8] A. J. Howse, H. De Sterck, R. D. Falgout, S. MacLachlan, and J. Schroder. Parallel-in- time multigrid with adaptive spatial coarsening for the linear advection and inviscid Burgers equations. SIAM Journal on Scientific Computing, 41:A538–A565, 2019. [9] X. Deng, X.-C. Cai, and J. Zou. A parallel space–time domain decomposition method for unsteady source inversion problems. Inverse Problems and Imaging, 9:1069–1091, 2015. [10] X. Deng, X.-C. Cai, and J. Zou. Two-level space–time domain decomposition methods for three-dimensional unsteady inverse source problems. Journal of Scientific Computing, 67:860–882, 2016. [11] A. T. Barker and M. Stoll. Domain decomposition in time for PDE-constrained optimization. Computer Physics Communications, 197:136–143, 2015. [12] H. Yang and X.-C. Cai. Two-level space–time domain decomposition methods for flow control problems. Journal of Scientific Computing, 70:717–743, 2017. [13] T.-T.-P. Hoang, C. Japhet, M. Kern, and J.E. Roberts. Space-time domain decomposition for reduced fracture models in mixed formulation. SIAM Journal on Numerical Analysis, 54:288–316, 2016. [14] M. Zheng, F. Liu, I. Turner, and V. Anh. A novel high order space-time spectral method for the time fractional Fokker–Planck equation. SIAM Journal on Scientific Computing, 37:A701–A724, 2015. [15] H. Yang and F.-N. Hwang. A nonlinear elimination preconditioner for fully coupled space-time solution algorithm with applications to high-Rayleigh number thermal convective flow problems. Communications in Computational Physics, 26:749–767, 2019. [16] S. Li and X.-C. Cai. Convergence analysis of two-level space-time additive Schwarz method for parabolic equations. SIAM Journal on Numerical Analysis, 53:2727– 2751, 2015. [17] X.-C. Cai and D. E. Keyes. Nonlinearly preconditioned inexact Newton algorithms. SIAM Journal on Scientific Computing, 24:183–200, 2002. [18] F.-N. Hwang and X.-C. Cai. Improving robustness and parallel scalability of Newton method through nonlinear preconditioning. In Lecture Notes in Computational Science and Engineering, pages 201–208. Springer-Verlag, Heidelberg, 2004. [19] L. Liu, D. E. Keyes, and S. Sun . Fully implicit two-phase reservoir simulation with the additive Schwarz preconditioned inexact Newton method. In SPE Reservoir Characterization and Simulation Conference and Exhibition. Society of Petroleum Engineers, 2013. [20] H.-B. An. On convergence of the additive Schwarz preconditioned inexact Newton method. SIAM Journal on Numerical Analysis, 43:1850–1871, 2005. [21] S. H. Lui. Nonlinearly preconditioned Newton’s method. In Proceedings of the 14th International Conference on Domain Decomposition Methods, Cocoyoc, Mexico, pages 95–105. Citeseer, 2002. [22] C. Groß and R. Krause. On the globalization of ASPIN employing trust-region control strategies–convergence analysis and numerical examples. arXiv preprint arXiv:2104.05672, 2021. [23] O. Moyner Ø. Klemetsdal, A. Moncorgé and K. Lie. Additive Schwarz preconditioned exact Newton method as a nonlinear preconditioner for multiphase porous media flow. In ECMOR XVII, volume 2020, pages 1–20. European Association of Geoscientists & Engineers, 2020. [24] O. Møyner, A. F. Rasmussen, Ø. Klemetsdal, H. M. Nilsen, A. Moncorgé, and K.- A. Lie. Nonlinear domain-decomposition preconditioning for robust and efficient field-scale simulation of subsurface flow. Computational Geosciences, 28:241–251, 2024. [25] Ø. Klemetsdal, A. Moncorgé, O. Møyner, and K.-A. Lie. A numerical study of the additive Schwarz preconditioned exact Newton method (ASPEN) as a nonlinear preconditioner for immiscible and compositional porous media flow. Computational Geosciences, 26:1045–1063, 2022. [26] L. Liu and D. E. Keyes. Field-split preconditioned inexact Newton algorithms. SIAM Journal on Scientific Computing, 37:A1388–A1409, 2015. [27] L. Liu and D. E. Keyes. Convergence analysis for the multiplicative Schwarz preconditioned inexact Newton algorithm. SIAM Journal on Numerical Analysis, 54:3145– 3166, 2016. [28] H. Yu L. Liu, W. Gao and D. E. Keyes. Overlapping multiplicative Schwarz preconditioning for linear and nonlinear systems. Journal of Computational Physics, 496:112548, 2024. [29] L. Liu, W. Zhang, and D. E. Keyes. Nonlinear multiplicative Schwarz preconditioning in natural convection cavity flow. In Domain Decomposition Methods in Science and Engineering XXIII, pages 227–235. Springer, 2017. [30] V. Dolean, M. J. Gander, W. Kheriji, F. Kwok, and R. Masson. Nonlinear preconditioning: How to use a nonlinear Schwarz method to precondition Newton’s method. SIAM Journal on Scientific Computing, 38:A3357–A3380, 2016. [31] P. M. Kumbhar F. Chaouqui, M. J. Gander and T. Vanzan. Linear and nonlinear substructured restricted additive Schwarz iterations and preconditioning. Numerical Algorithms, 91:81–107, 2022. [32] F. Chaouqui, M. J. Gander, P. M. Kumbhar, and T. Vanzan. On the nonlinear Dirichlet–Neumann method and preconditioner for Newton’s method. In Domain Decomposition Methods in Science and Engineering XXVI, pages 381–389. Springer, 2023. [33] G. Ciaramella and L. Mechelli. An overlapping waveform relaxation preconditioner for economic optimal control problems with state constraints. In Domain Decomposition Methods in Science and Engineering XXVI, pages 461–469. Springer, 2023. [34] F.-N. Hwang and X.-C. Cai. A combined linear and nonlinear preconditioning technique for incompressible Navier-Stokes equations. In International Workshop on Applied Parallel Computing, pages 313–322. Springer, 2004. [35] X.-C. Cai, D. E. Keyes, and L. Marcinkowski. Non-linear additive Schwarz preconditioners and application in computational fluid dynamics. International Journal for Numerical Methods in Fluids, 40:1463–1470, 2002. [36] L. Marcinkowski and X.-C. Cai. Parallel performance of some two-level ASPIN algorithms. In Domain Decomposition Methods in Science and Engineering, pages 639–646. Springer, 2005. [37] F.-N. Hwang and X.-C. Cai. A class of parallel two-level nonlinear Schwarz preconditioned inexact Newton algorithms. Computer Methods in Applied Mechanics and Engineering, 196:1603–1611, 2007. [38] X.-C. Cai, L. Marcinkowski, and P. S. Vassilevski. An element agglomeration nonlinear additive Schwarz preconditioned Newton method for unstructured finite element problems. Applications of Mathematics, 50:247–275, 2005. [39] G. Ciaramella, F. Kwok, and G. Müller. A nonlinear optimized Schwarz preconditioner for elliptic optimal control problems. In Domain Decomposition Methods in Science and Engineering XXVI, pages 391–398. Springer, 2023. [40] Z. Y. Wong, F. Kwok, R. N. Horne, and H. A. Tchelepi. Sequential-implicit Newton method for multiphysics simulation. Journal of Computational Physics, 391:155–178, 2019. [41] E. Keilegavlen, J. O. Skogestad, and J. M. Nordbotten. Domain decomposition preconditioning for non-linear elasticity problems. In Proceedings of 11th World Congress of Computational Mechanics. International Centre for Numerical Methods in Engineering, 2014. [42] H. De Sterck and M. Winlaw. A nonlinearly preconditioned conjugate gradient algorithm for rank-R canonical tensor approximation. Numerical Linear Algebra with Applications, 22:410–432, 2015. [43] C. Andrea, F. Giovanni, and R. Massimo. Preconditioning strategies for nonlinear conjugate gradient methods, based on quasi-Newton updates. In AIP conference proceedings, volume 1776. AIP Publishing, 2016. [44] M. Al-Baali, A. Caliciotti, G. Fasano, and M. Roma. Quasi-Newton based preconditioning and damped quasi-Newton schemes for nonlinear conjugate gradient methods. Numerical Analysis and Optimization, pages 1–21. Springer, 2018. [45] A. Caliciotti, G. Fasano, and M. Roma. Preconditioned nonlinear conjugate gradient methods based on a modified secant equation. Applied Mathematics and Computation, 318:196–214, 2018. [46] H. De Sterck and A. J. Howse. Nonlinearly preconditioned L-BFGS as an acceleration mechanism for alternating least squares with application to tensor decomposition. Numerical Linear Algebra with Applications, 25:e2202, 2018. [47] E. Acar, D. M. Dunlavy, and T. G. Kolda. A scalable optimization approach for fitting canonical tensor decompositions. Journal of Chemometrics, 25:67–86, 2011. [48] H. De Sterck. A nonlinear GMRES optimization algorithm for canonical tensor decomposition. SIAM Journal on Scientific Computing, 34:A1351–A1379, 2012. [49] H. De Sterck. Steepest descent preconditioning for nonlinear GMRES optimization. Numerical Linear Algebra with Applications, 20:453–471, 2013. [50] H. De Sterck and A. Howse. Nonlinearly preconditioned optimization on grassmann manifolds for computing approximate Tucker tensor decompositions. SIAM Journal on Scientific Computing, 38:A997–A1018, 2016. [51] H. De Sterck, Y. He, and O. A. Krzysik. Anderson acceleration as a Krylov method with application to asymptotic convergence analysis. arXiv preprint arXiv:2109.14181, 2021. [52] D. Wang, Y. He, and H. De Sterck. On the asymptotic linear convergence speed of Anderson acceleration applied to ADMM. Journal of Scientific Computing, 88:38, 2021. [53] H. De Sterck and Y. He. On the asymptotic linear convergence speed of Anderson acceleration, Nesterov acceleration, and nonlinear GMRES. SIAM Journal on Scientific Computing, 43:S21–S46, 2021. [54] S. Wang, P. H. Winterfeld, and Y.-S. Wu. An efficient adaptive nonlinearity elimination preconditioned inexact Newton method for parallel simulation of thermalhydraulic- mechanical processes in fractured reservoirs. SPE Reservoir Simulation Conference, page D011S002R003. SPE, 2015. [55] X.-C. Cai and X. Li. Inexact Newton methods with restricted additive Schwarz based nonlinear elimination for problems with high local nonlinearity. SIAM Journal on Scientific Computing, 33:746–762, 2011. [56] L. Luo, R. Chen, X.-C. Cai, and D. E. Keyes. A nonlinear elimination preconditioned inexact Newton algorithm for steady state incompressible flow problems on 3D unstructured meshes. In Domain Decomposition Methods in Science and Engineering XXV, pages 441–449. Springer, 2020. [57] H. Yang and F.-N. Hwang. An adaptive nonlinear elimination preconditioned inexact Newton algorithm for highly local nonlinear multicomponent PDE systems. Applied Numerical Mathematics, 133:100–115, 2018. [58] S.-R. Cai and F.-N. Hwang. A hybrid-line-and-curve search globalization technique for inexact Newton methods. Applied Numerical Mathematics, 173:79–93, 2022. [59] P. Gosselet, C. Negrello, and C. Rey. Nonlinearly preconditioned FETI method. In MAFELAP, 2019. [60] A. Klawonn, M. Lanser, and O. Rheinbach. Nonlinear FETI-DP and BDDC methods. SIAM Journal on Scientific Computing, 36:A737–A765, 2014. [61] A. Klawonn, M. Lanser, and M. Uran. Adaptive nonlinear elimination in nonlinear FETI-DP methods. In Domain Decomposition Methods in Science and Engineering XXVI, pages 337–345. Springer International Publishing, 2022. [62] C. Negrello, P. Gosselet, and C. Rey. Nonlinearly preconditioned FETI solver for substructured formulations of nonlinear problems. Mathematics, 9:3165, 2021. [63] A. Heinlein, A. Klawonn, and M. Lanser. Adaptive nonlinear domain decomposition methods with an application to the p-Laplacian. SIAM Journal on Scientific Computing, 45:S152–S172, 2022. [64] A. Klawonn, M. Lanser, and O. Rheinbach. Nonlinear BDDC methods with approximate solvers. Electronic Transactions on Numerical Analysis, 49:244–273, 2018. [65] A. Klawonn, M. Lanser, O. Rheinbach, and M. Uran. Nonlinear FETI-DP and BDDC methods: a unified framework and parallel results. SIAM Journal on Scientific Computing, 39:C417–C451, 2017. [66] H. Yang, S. Sun, and C. Yang. Nonlinearly preconditioned semismooth Newton methods for variational inequality solution of two-phase flow in porous media. Journal of Computational Physics, 332:1–20, 2017. [67] F.-N. Hwang, H.-L. Lin, and X.-C. Cai. Two-level nonlinear elimination based preconditioners for inexact Newton methods with application in shocked duct flow calculation. Electronic Transactions on Numerical Analysis, 37:239–251, 2010. [68] F.-N. Hwang, Y.-C. Su, and X.-C. Cai. A parallel adaptive nonlinear elimination preconditioned inexact Newton method for transonic full potential equation. Computers and Fluids, 110:96–107, 2015. [69] H. Tang, S. Wang, C. Yin, Y. Di, Y.-S. Wu, and Y. Wang. Fully-coupled multiphysical simulation with physics-based nonlinearity-elimination preconditioned inexact Newton method for enhanced oil recovery. Communications in Computational Physics, 1:244–265, 2019. [70] L. Luo, X.-C. Cai, Z. Yan, L. Xu, and D. E. Keyes. A multilayer nonlinear elimination preconditioned inexact Newton method for steady-state incompressible flow problems in three dimensions. SIAM Journal on Scientific Computing, 42:B1404– B1428, 2020. [71] H. Yang, Y. Li, and S. Sun. Nonlinearly preconditioned constraint-preserving algorithms for subsurface three-phase flow with capillarity. Computer Methods in Applied Mechanics and Engineering, 367:113140, 2020. [72] H. Yang, S. Sun, Y. Li, and C. Yang. Parallel reservoir simulators for fully implicit complementarity formulation of multicomponent compressible flows. Computer Physics Communications, 244:2–12, 2019. [73] L. Luo, X.-C. Cai, and D. E. Keyes. Nonlinear preconditioning strategies for two-phase flows in porous media discretized by a fully implicit discontinuous Galerkin method. SIAM Journal on Scientific Computing, 43:S317–S344, 2021. [74] L. Luo, W.-S. Shiu, R. Chen, and X.-C. Cai. A nonlinear elimination preconditioned inexact Newton method for blood flow problems in human artery with stenosis. Journal of Computational Physics, 399:108926, 2019. [75] S. Gong and X.-C. Cai. A nonlinear elimination preconditioned inexact Newton method for heterogeneous hyperelasticity. SIAM Journal on Scientific Computing, 41:S390–S408, 2019. [76] S. Gong and X.-C. Cai. A nonlinear elimination preconditioned Newton method with applications in arterial wall simulation. In Domain Decomposition Methods in Science and Engineering XXIV, pages 353–361. Springer, 2018. [77] J. Huang, C. Yang, and X.-C. Cai. A nonlinearly preconditioned inexact Newton algorithm for steady Lattice Boltzmann equations. SIAM Journal on Scientific Computing, 38:A1701–A1724, 2016. [78] T. A. Brunner, T. S. Haut, and P. F. Nowak. Nonlinear elimination applied to radiation diffusion. Nuclear Science and Engineering, 194:939–951, 2020. [79] L. Liu, F.-N. Hwang, L. Luo, X.-C. Cai, and D. E. Keyes. A nonlinear elimination preconditioned inexact Newton algorithm. SIAM Journal on Scientific Computing, 44:A1579–A1605, 2022. [80] L. Luo and X.-C. Cai. Preconditioned inexact Newton with learning capability for nonlinear system of equations. SIAM Journal on Scientific Computing, 45:A849– A871, 2023. [81] C. Hu. On widening region of convergence for Newton’s method–a numerical ODE perspective. International Journal of Computer Mathematics, 78:433–444, 2001. [82] N. C. Nguyen, P. Fernandez, R. M. Freund, and J. Peraire. Accelerated residual methods for the iterative solution of systems of equations. SIAM Journal on Scientific Computing, 40:A3157–A3179, 2018. [83] R. J. LeVeque. Numerical Methods for Conservation Laws. Birkhauser, Basel, 1992. [84] D. P. Ballou. Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions. Transactions of the American Mathematical Society, 152:441–460, 1970. [85] S. Marx, T. Weisser, D. Henrion, and J. Lasserre. A moment approach for entropy solutions to nonlinear hyperbolic PDEs. arXiv preprint arXiv:1807.02306, 2018. [86] L. Schneider, R. di Chiara Roupert, G. Schäfer, and P. Helluy. Highly gravity-driven flow of a NAPL in water-saturated porous media using the discontinuous Galerkin finite-element method with a generalised Godunov scheme: Highly gravity-driven flow of a NAPL in water-saturated porous media. Computational Geosciences, 19:855–876, 2015. [87] P. H. Iryanto, Gunawan. An OpenMP parallel Godunov scheme for 1D two phase oil displacement problem. In 2017 5th International Conference on Information and Communication Technology (ICoIC7), pages 1–5. IEEE, 2017. [88] C. W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes, ii. Journal of Computational Physics, 83:32–78, 1989. [89] R. S. Dembo, S. C. Eisenstat, and T. Steihaug. Inexact Newton methods. SIAM Journal on Numerical Analysis, 19:400–408, 1982. [90] L. Armijo. Minimization of functions having Lipschitz continuous first partial derivatives. Pacific Journal of Mathematics, 16:1–3, 1966. [91] H. Yang, F.-N. Hwang, and X.-C. Cai. Nonlinear preconditioning techniques for fullspace Lagrange-Newton solution of PDE-constrained optimization problems. SIAM Journal on Scientific Computing, 5:A2756–A2778, 2016. [92] K. Jittorntrum. An implicit function theorem. Journal of Optimization Theory and Applications, 25:575–577, 1978. [93] Y. Saad and M. H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7:856–869, 1986. [94] Y. Zhu and A. H. Sameh. How to generate effective block Jacobi preconditioners for solving large sparse linear systems. Advances in Computational Fluid-Structure Interaction and Flow Simulation: New Methods and Challenging Computations, pages 231–244, 2016. [95] M. Rathinam, L. R. Petzold, Y. Cao, and D. T. Gillespie. Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method. The Journal of Chemical Physics, 119:12784–12794, 2003. [96] L. J. Pyrak-Nolte and J. P. Morris. Single fractures under normal stress: The relation between fracture specific stiffness and fluid flow. International Journal of Rock Mechanics and Mining Sciences, 37:245–262, 2000. [97] S.-H. Weng, Q. Chen, N. Wong, and C.-K. Cheng. Circuit simulation via matrix exponential method for stiffness handling and parallel processing. In Proceedings of the International Conference on Computer-Aided Design, pages 407–414, 2012. [98] C. F. Curtiss and J. O. Hirschfelder. Integration of stiff equations. Proceedings of the National Academy of Sciences, 38:235–243, 1952. [99] C. W. Gear. The automatic integration of ordinary differential equations. Communications of the ACM, 14:176–179, 1971. [100] J. G. Verwer. An analysis of Rosenbrock methods for nonlinear stiff initial value problems. SIAM Journal on Numerical Analysis, 19:155–170, 1982. [101] L. F. Shampine and M. W. Reichelt. The MATLAB ODE suite. SIAM Journal on Scientific Computing, 18:1–22, 1997. [102] C. B. Moler and G. W. Stewart. An algorithm for generalized matrix eigenvalue problems. SIAM Journal on Numerical Analysis, 10:241–256, 1973. [103] G. W. Stewart. On the sensitivity of the eigenvalue problem Ax = λBx. SIAM Journal on Numerical Analysis, 9:669–686, 1972. [104] G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, 4th edition, 2013. [105] R. A. Jabr, M. Hamad, and Y. M. Mohanna. Newton-Raphson solution of Poisson’s equation in a pn diode. International Journal of Electrical Engineering Education, 44:23–33, 2007. [106] D. P. Young, R. G. Melvin, M. B. Bieterman, F. T. Johnson, S. S. Samant, and J. E. Bussoletti. A locally refined rectangular grid finite element method: application to computational fluid dynamics and computational physics. Journal of Computational Physics, 92:1–66, 1991. [107] W. Gander and G. H. Golub. Cyclic reduction-history and applications. In Scientific Computing: Proceedings of the Workshop, 10-12 March 1997, Hong Kong, page 73. Springer Science & Business Media, 1998. [108] D. I. Ketcheson, R. J. LeVeque, and M. J. del Razo. Riemann Problems and Jupyter Solutions. SIAM, Philadelphia, 2020. [109] C. B. Laney. Computational Gasdynamics. Cambridge University Press, Cambridge, 1998. [110] O. Castro-Orgaz and W. H. Hager. Shallow Water Hydraulics. Springer, New York, 2019. [111] F. Benkhaldoun and M. Seaïd. A simple finite volume method for the shallow water equations. Journal of Computational and Applied Mathematics, 234:58–72, 2010. [112] P. L. Roe. Characteristic-based schemes for the Euler equations. Annual Review of fluid Mechanics, 18:337–365, 1986.
指導教授	黃楓南(Feng-Nan Hwang)	審核日期	2024-7-26
推文	facebook   plurk   twitter   funp   google   live   udn   HD   myshare   reddit   netvibes   friend   youpush   delicious   baidu
網路書籤	Google bookmarks   del.icio.us   hemidemi   myshare