Multiblock Method for Fluid Flow
Concepts, Algorithms, and Applications
- 2026
- Book
- Author
- Hansong Tang
- Book Series
- Nonlinear Systems and Complexity
- Publisher
- Springer Nature Switzerland
About this book
This book presents the multiblock method, also known by other names such as the zonal method and the domain decomposition method. The multiblock method offers a systematic approach to tackle large-scale, intricate problems by breaking them down into smaller, more manageable sub-problems. The method addresses each sub-problem individually while accounting for its interconnections with the others, ultimately arriving at a comprehensive solution. The book provides a cohesive overview of the multiblock method’s concepts and principles, particularly in the context of fluid flows, encompassing diverse fields including computational science, aerospace engineering, civil engineering, physical oceanography, and machine learning. It delves into foundational mathematics, studies model problems, elucidates numerical algorithms, and offers practical examples relevant to fluid dynamics. With its comprehensive coverage, this book serves as a resource for both learners and practitioners, catering to students, researchers, and modelers alike, whether as a textbook for structured learning or as a reference for applied problem-solving.
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Table of Contents
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Frontmatter
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Chapter 1. Fundamentals of Domain Decomposition
Hansong TangThis chapter delves into the fundamentals of domain decomposition methods, with a particular focus on the Schwarz method. It begins by introducing the concept of domain decomposition, tracing its origins back to the work of German mathematician Hermann Amandus Schwarz. The chapter explores the Schwarz method, including its alternating and parallel variants, and discusses the use of overlapping and patched subdomains. It also covers the application of domain decomposition to time-dependent problems, presenting both the conventional Schwarz method and the Schwarz waveform relaxation method. The chapter further extends to domain decomposition at the discrete level, discussing the multiplicative and additive Schwarz methods, and the Schur complement system. The text provides a comprehensive overview of these methods, their underlying principles, and their practical applications, making it an essential resource for professionals in the field of computational mathematics and engineering.AI Generated
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AbstractThe multiblock method traces back to the idea of domain decomposition proposed by the German mathematician Schwarz in 1869. Now, domain decomposition has evolved into a powerful, indispensable approach for numerically solving partial differential equations of fluid flows. This chapter presents the domain decomposition method in a relatively straightforward manner, making it understandable without advanced mathematical knowledge. The chapter starts with the classic Schwarz method for a boundary value problem of the Laplace equation. It introduces basic concepts, including subdomains, interfaces, and interface/transmission conditions, followed by a discussion on the convergence of the method. Then, this chapter proceeds to the computation of a time-dependent problem, and it presents the frameworks of two methods to be followed in the subsequent chapters: the conventional Schwarz method and the Schwarz waveform relaxation method. Furthermore, the discussion proceeds to domain decomposition at the discrete level, now a foundation for parallel computation. It presents relevant techniques, such as the multiplicative Schwarz method, the additive Schwarz method, and the Schur complement system. -
Chapter 2. Advection–Diffusion–Reaction Equation
Hansong TangThis chapter presents a study on the computation of coupled advection-diffusion-reaction equations, focusing on the multiblock method and its frameworks for simulation problems. It begins by introducing the computational problem and then presents the frameworks of the conventional Schwarz method and the Schwarz waveform relaxation method. The chapter provides a detailed discussion of the two methods, including their discretized equations, boundary conditions, and discretization accuracy. For the conventional Schwarz method, the chapter begins with an explicit scheme that utilizes the Dirichlet-Dirichlet interface condition. It then discusses the computation of an implicit discretization and implicit interface treatment using an iterative method, applying the Scarborough criterion as a condition for convergence. When a direct method (i.e., matrix inversion) is employed, an expression for the contraction factor is derived and numerically validated. Building on this, an optimized interface algorithm is introduced to accelerate convergence, achieving 'perfect convergence'—convergence within two iterations. Numerical examples validate the derived contraction factor and the perfect convergence. Additionally, the theoretical analysis is extended to nonlinear equations, with supporting numerical experiments confirming its validity. For the computation using the waveform relaxation Schwarz method, the governing PDEs are semi-discretized in space, resulting in a system of ordinary differential equations. These equations are then transformed into a system of algebraic equations via the Fourier transform, and the computation is reformulated as a Schwarz iteration of the system. For the classical interface algorithm, the contraction factor is derived, demonstrating that the iteration converges. A contraction factor is also derived for the case with an optimized interface algorithm. Because the analysis is conducted on the s-domain, the parameters in the optimized algorithm are approximated in the t-domain for the actual computation. Moreover, the analysis and derivation for the contraction factor are extended to the nonlinear problems. Numerical experiments illustrate and verify the theoretical results. A comparison is made between the conventional method and the waveform relaxation method in terms of computational time, concluding that the waveform relaxation method requires at least three times as many computation units as the conventional method, for both classical and optimized interface algorithms. The chapter concludes with a summary and a set of questions to aid in understanding, discussing, and investigating the content.AI Generated
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AbstractThis chapter analyzes the computation of advection–diffusion–reaction equations via the domain decomposition method. The equations in subdomains may differ, e.g., a diffusion equation in a subdomain and an advection–diffusion in another, and they are coupled at a subdomain interface. It deals with two computation methods: the conventional Schwarz method and the Schwarz waveform relaxation method. For the conventional Schwarz method, the chapter discusses the computation by explicit and implicit schemes at the full discretization level. Expressions are derived for the convergence speed associated with the classical interface algorithm based on the Dirichlet condition, and then with its optimized interface algorithm to accelerate the convergence. The optimized interface algorithm leads to “perfect convergence,” that is, convergence within two iterations. The classical and optimized algorithms also extend to nonlinear equations, and numerical examples illustrate their validity. For the Schwarz waveform relaxation method, this chapter discusses the computation at the semi-discretization level, starting from linear equations and then proceeding to nonlinear equations. It presents the expressions of convergence speeds with the classical interface algorithm and also the optimized interface algorithm that effectively speeds up the convergence of computation. Finally, the chapter compares the computational loads of the conventional method and the waveform relaxation method, showing that the former method has a considerably lower load. -
Chapter 3. Compressible Flow
Hansong TangThis chapter begins with conservation laws and their numerical methods, which form the foundation for simulating compressible flows. It introduces fundamental concepts, including hyperbolic systems, weak solutions, and piecewise smooth solutions. The theory states that a piecewise smooth solution is a weak solution under certain conditions, and such a solution satisfies a conservation condition. Conservative numerical schemes are then introduced, accompanied by typical examples. A theorem shows that the converged solution of a conservative scheme is a weak solution. Additionally, it is demonstrated that such a solution satisfies a discrete form of the conservation condition. The chapter then proceeds to numerical computations using the multiblock method, with a particular focus on computations involving patched and overlapping grids, especially at their interfaces. Interface algorithms are divided into two categories: conservative and nonconservative algorithms. For both patched and overlapping grids, conservative interface algorithms based on enforcing numerical fluxes are presented. Nonconservative interface algorithms built on interpolation are also discussed. The setup of initial values for the computations is specified. When a conservative interface algorithm is applied, it is shown that the numerical solution satisfies the conservation condition and the converged solution is a weak solution. These theoretical results are supported by numerical experiments, which demonstrate the advantages of conservative interface algorithms over nonconservative ones. In particular, while a nonconservative treatment may introduce errors in the strength and location of discontinuities or even yield a non-physical or incorrect solution, conservative treatments avoid such issues. However, conservative interface algorithms, particularly those based on enforcing numerical fluxes, are not without challenges. First, analysis shows that when two subdomains have the same grid spacing but use different numerical schemes, such a interface algorithm may become inconsistent with the PDE, effectively computing a different equation at the interface. Second, when two subdomains use the same numerical scheme but differ in grid spacing, a similar inconsistency may arise. Numerical experiments reveal that such inconsistencies can lead to artifacts at grid interfaces, erroneous solutions, and even the breakdown of the computational process. As a remedy, numerical fluxes that are independent of grid spacing and time step, such as those used in the Godunov scheme, can mitigate the issue. Finally, the chapter turns to nonconservative interface treatments. It introduces the concept of conservation error and analyzes its magnitude in numerical computations. A theoretical foundation is provided for nonconservative treatments, proving that a converged solution can still be a weak solution under two specific conditions. These conditions may be verified numerically. Corresponding experiments are presented to validate the performance of nonconservative interface algorithms under these conditions. Additionally, counterexamples are provided to demonstrate that when these conditions are not met, the resulting solution becomes invalid.AI Generated
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AbstractThis chapter starts with conservation laws and their numerical methods, which are the bases for the simulation of compressible flows. It presents concepts such as weak solution and conservation error, and discusses classical conservative schemes. Then, it proceeds to compute the conservation laws via domain decomposition. It discusses conservative interface algorithms and proves that a converged numerical solution is a weak solution of a conservation law. Numerical examples demonstrate the advantages of conservative interface algorithms in correctly capturing shock location and strength. Analysis indicates that the conservative interface algorithm built by directly enforcing numerical fluxes that contain grid spacing and/or time step suffers from problems in two scenarios: it becomes inconsistent with the conservation law when the subdomains have different grid spacings and when they use different discretization schemes. Following these, the chapter deals with the computation with nonconservative interface algorithms. It analyzes the conservation error resulting from a nonconservative interface treatment. Theoretical analysis explains why a nonconservative interface algorithm may work; it proves that the converged solution associated with the nonconservative interface algorithm is a weak solution under certain conditions, followed by illustrative numerical examples. Additionally, a counterexample illustrates that a nonconservative interface algorithm can fail. -
Chapter 4. Incompressible Flow
Hansong TangThis chapter explores the multiblock method for simulating incompressible flows, focusing on the governing equations, solution techniques, and interface algorithms. It begins with an overview of the continuity, momentum, and energy equations that describe incompressible fluid flow. The chapter then delves into the discretization and solution methods, including the use of a second-order accurate, implicit finite difference method and the artificial compressibility (AC) method for handling the divergence-free condition. A significant portion of the text is dedicated to the mass-flux-balance interpolation (MFBI) algorithm, which is compared with standard interpolation (SI) methods. The advantages of MFBI, such as reduced discrepancies and oscillations in solutions, are highlighted through numerical experiments on impulsive and oscillatory cavity flows, as well as pipe-bend and cylinder flows. The chapter concludes with a discussion on the performance of the multiblock method and the MFBI algorithm, demonstrating their effectiveness in capturing complex flow patterns and ensuring mass conservation. This detailed analysis provides valuable insights into the application of advanced computational techniques in fluid dynamics.AI Generated
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AbstractThis chapter discusses the simulation of incompressible flows by computing the Navier–Stokes equations via the multiblock method. The Navier–Stokes equations in curvilinear coordinates are discretized using a second-order finite difference method on overset/composite grids, and the discretized equations are solved by the artificial compressible method. The computation utilizes the conventional Schwarz method and other techniques, including implicit residual smoothing, local dual-time stepping, and V-cycle multigrid acceleration. Grid connectivity and interface treatment are discussed. An interface method is derived to enforce mass conservation at grid interfaces, which is crucial in the simulation of an incompressible flow. The method is referred to as mass flux balance interpolation (MFBI); it is a slight modification of the standard interpolation (SI) at the interfaces, and it is straightforward to implement. Numerical experiments are conducted in cavity flow, pipe-bend flow, and flow past a cylinder to test the multiblock method. These experiments demonstrate that, compared with the SI, the MFBI removes or suppresses numerical oscillations at grid interfaces, leads to faster convergence of the Schwarz iteration, and tends to reduce conservation errors of numerical solutions. -
Chapter 5. Coastal Ocean Flow
Hansong TangThe chapter begins by highlighting the need for advanced modeling capabilities to simulate emerging, complex ocean flows that involve distinct types of physics across vast temporal and spatial scales. It introduces a multiblock model system that couples the Navier-Stokes (NS) equations for near-field flows with the Geophysical Fluid Dynamics (GFD) equations for far-field flows. The chapter delves into the governing equations of the NS-GFD model system, the solvers used for the NS and GFD equations, and the techniques for coupling these equations, including grid connectivity, interface algorithms, and computational procedures. A series of numerical examples, such as the sill flow, flow past bridge piers, and thermal effluent discharge, are presented to demonstrate the performance and accuracy of the NS-GFD model system. The chapter also explores the effectiveness of a pressure-splitting technique in mitigating numerical artifacts and non-physical solutions. Additionally, it introduces interface condition II, which enforces momentum continuity across interfaces, and compares it with interface condition I through numerical simulations of riverbend flows. The chapter then addresses the coupling of the Shallow Water (SW) equations with the GFD equations to capture surface wave dynamics, presenting numerical examples of tidal flow and dam-break flow to illustrate the performance of the SW-GFD model system. The chapter concludes by discussing the potential applications of these model systems and the need for further testing and development of conservative interface treatments.AI Generated
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AbstractThe state-of-the-art engineering and ocean science models are based on the Navier–Stokes (NS) equations and the geophysical fluid dynamics (GFD) equations, respectively. This chapter presents a multiblock modeling system that couples the NS equations and the GFD equations in two ways to simulate emerging multiscale, multiphysics ocean flows, especially local, complex phenomena. In the NS-GFD model system, the NS equations are solved by a finite difference method, and the GFD equations are solved by a finite volume method. The model system extends the framework and the overset-grid techniques, including the grid-connectivity and interface algorithms, of the previous chapters, and it also adopts a pressure split technique to couple the NS and GFD equations. Numerical examples, including a sill flow, a current past bridge piers, and a thermal-effluent discharge into a realistic ocean, illustrate the promise of the NS-GFD model system. However, difficulties such as non-physical phenomena may occur in numerical solutions, e.g., at NS-GFD interfaces. Then, this chapter discusses methods to overcome these difficulties, explores interface conditions, and includes example simulations to demonstrate their performance. Additionally, it extends the framework of the NS-GFD model system to couple the shallow water equations with the GFD equations and presents numerical experiments on a surface tide, a dam-break flow, and a coastal flood. -
Chapter 6. Special Topics
Hansong TangThis chapter delves into the phenomenon of odd-even oscillation in numerical solutions of differential equations, particularly in fluid dynamics. It begins by defining the concept and investigating the oscillation in the numerical solution of a model problem on a single domain, then extends the analysis to two subdomains. Two types of odd-even oscillation are considered: dual-mode-solution odd-even oscillation and inconsistent-boundary-conditions odd-even oscillation. Necessary and sufficient conditions for the onset of each type are derived and illustrated with numerical examples. The chapter also explores a novel idea: solving PDEs coupled at their subdomain interface via ML solutions at the interface. This idea is motivated by the limitations of the classical coupling, the success of ML in applications, and the potential of the ML coupling in terms of capability, efficiency, and universality. The chapter formulates the ML coupling using the concepts of interface zones and ML models for local solutions. The discussion begins with the ML coupling for a boundary value problem of the Poisson equation, illustrating it with numerical experiments. It then extends the coupling to advection-diffusion-reaction equations. The error in the ML-coupled solutions is analyzed and attributed to two sources: errors inherited in the training data and errors caused by the ML model itself. The concept of an ML Godunov scheme is introduced and applied to the ML coupling. Numerical experiments indicate that, after training, ML models exhibit some predictive capability beyond merely reproducing the training data. As a step toward fluid flow simulation, machine learning is used to simulate the cavity flow and generate the flow solution within an interface zone. Both topics, odd-even oscillation and the ML coupling, warrant further study. The analysis of odd-even oscillation is based on a simplified model and does not yet directly explain more complex phenomena. The ML coupling approach has shown promise in solving PDEs, but its full potential requires additional exploration, particularly with respect to the proposed ML Godunov scheme and its effectiveness in coupling. A more systematic investigation is needed to establish the theoretical foundations and test the practical potential of the ML coupling in delivering the conjectured capabilities, efficiency, and universality, especially for fluid flows.AI Generated
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AbstractThis chapter discusses two topics related to computation using the multiblock method. The first topic is the artificial odd–even oscillation that frequently occurs in numerical solutions. The chapter starts with the computation of a model equation on a single grid. It identifies the odd–even oscillation resulting from two causes: a dual-mode solution and inconsistent boundary conditions. The chapter presents criteria for the onset of the oscillation, followed by their illustration through numerical experiments. Then, the chapter proceeds to the computation on two grids, presenting criteria for its onset and numerical examples. Finally, it discusses the odd–even oscillation associated with more complicated equations.The second topic concerns coupling numerical solutions of differential equations at subdomain interfaces by machine learning (ML), representing a potentially novel paradigm and an exploratory study. The basic idea of ML coupling is to utilize ML models to compute subdomain interface conditions. This chapter outlines the concepts, approaches, and algorithms for the ML coupling. It begins with a boundary value problem of the Poisson equation. Then, the chapter extends the coupling to an initial value problem of parabolic equations and proposes an ML Godunov method to conduct the ML coupling. Numerical examples demonstrate the promising performance of the ML coupling. To further illustrate the promise, a preliminary numerical result is presented for an ML solution in the interface zone of a cavity flow. -
Backmatter
- Title
- Multiblock Method for Fluid Flow
- Author
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Hansong Tang
- Copyright Year
- 2026
- Publisher
- Springer Nature Switzerland
- Electronic ISBN
- 978-3-031-78568-9
- Print ISBN
- 978-3-031-78567-2
- DOI
- https://doi.org/10.1007/978-3-031-78568-9
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