Advances in Applied Mathematics and Mechanics
Advances in Applied Mathematics and Mechanics (AAMM) provides a fast communication platform among researchers using mathematics as a tool for solving problems in mechanics and engineering, with particular emphasis in the integration of theory and applications. To cover as wide audiences as possible, abstract or axiomatic mathematics is not encouraged. Innovative numerical analysis, numerical methods, and interdisciplinary applications are particularly welcome.
High-Order Numerical Approximation and Error Estimation for Fourth-Order Equation in Complex Cylindrical Domain
In this paper, we introduce a high-order numerical method for solving fourth-order equations in an elliptical cylindrical region. Initially, we employ an elliptical cylindrical coordinate transformation to reformulate the fourth-order equation as an equivalent second-order coupled system in the new coordinate system. To overcome the singularity introduced by coordinate transformation, an essential pole condition is derived. The weak form and its discretization are also established. Furthermore, the well-posedness of both the weak solution and its approximate solution have been theoretically investigated. By introducing novel projection operators and demonstrating their approximation properties, we prove the error estimates in conjunction with the approximation results of Fourier series. Finally, several numerical examples are provided to validate the convergence and high accuracy of our proposed schemes.
Copyright (c) 2025 Advances in Applied Mathematics and Mechanics
Analyzing Bending Problems of Plates on Elastic Foundations via Improved Element-Free Galerkin Method
In this study, we combine orthonormal basis functions with the traditional moving least-squares (MLS) approximation to establish a new approximation function via the improved moving least-squares (IMLS) approximation, and the corresponding formula derivation is provided in Section 2. Afterwards, the equilibrium, geometrical, and physical equations for bending problems of plates on elastic foundations are presented respectively, and the equivalent functional of such problems is established by imposing the essential boundary conditions via the penalty method. Subsequently, the calculation formulas for the numerical solution are derived using the improved element-free Galerkin (IEFG) method based on the IMLS approximation. In the numerical examples, we verify the convergence of the IEFG method by increasing the number of nodes. Compared with the EFG method, the IEFG method exhibits faster convergence. Furthermore, by adopting the IEFG method for solving three numerical examples, smaller errors and higher computational speed are achieved.
A Fully Discrete Method for Space-Time Fractional Parabolic Problems
This paper considers a parabolic equation involving the Caputo fractional time derivative of order $\gamma\in(0,1)$ and the spectral fractional power, of order $s\in(0,1)$, of a symmetric second order elliptic operator $\mathcal{L}$. By using the Caffarelli–Silvestre extension, the original problem is converted to a quasi-stationary elliptic problem on a semi-infinite cylinder in one more spatial dimension and with a dynamic boundary condition. Based on the solution representation with the Mittag–Leffler function and the eigenpairs of the elliptic operator $\mathcal{L}$, some new regularity results are derived. For the temporal semi-discrete scheme using the Alikhanov difference approximation, stability and convergence are established. Furthermore, the first-degree tensor product finite element method is adopted for the spatial discretization to obtain a fully discrete scheme, and an error estimate is derived. Finally, an efficient algorithm based on a generalized eigenvalue problem is applied to solve the matrix system of the full discretization. Numerical examples are provided to verify the theoretical results.
Space-time fractional parabolic equation
A Fast Spatial Two-Grid Raviart-Thomas MFE Algorithm for the Nonlinear Time Fractional Diffusion Equations on Temporal Graded Mesh
In this paper, a fast two-grid algorithm is constructed for solving the nonlinear time fractional diffusion equations by using the lowest order Raviart-Thomas mixed finite element (RTMFE) space and the temporal graded mesh. In the algorithm, the Caputo time fractional derivative is discretized by the well-known $L1$ formula on the graded mesh, the spatial domain is divided into coarse and fine grids, then a fast two steps algorithm is proposed by using two-grid computing method. The existence, uniqueness and unconditional stability for the proposed algorithm on the temporal graded mesh are derived in detail. In addition, when the analytical solution satisfies different regularity assumptions, the asymptotically optimal a priori error estimates in spatial direction are obtained on both the temporal uniform and graded meshes, which show that when spatial coarse and fine grid parameters satisfy $H=O(h^{1/2})$, the fast algorithm can obtain the same accuracy as the RTMFE algorithm. Finally, two numerical examples with different regularity conditions are provided to demonstrate the theoretical results.
nonlinear time fractional diffusion equations
Error Analyses of the Single Hidden Layer Neural Network Based on the Extreme Learning Machine for Solving Differential Equations
The differential equations, especially the stiff differential equations with the large eigenvalue and instability of solutions, impose strict limitations on step size for traditional numerical methods. The single hidden layer neural network method based on the extreme learning machine is widely used to solve various differential equations due to its advantages of few parameters and high efficiency, and the optimal estimation of parameters is obtained. In practice, this does not mean that the solutions of differential equations converge well. This paper aims to establish theoretical results for solving non-stiff and stiff differential equations using the single hidden layer neural network based on the extreme learning machine. Under the classical Lipschitz condition and the one-sided Lipschitz condition, we derive convergence results for the approximate solutions of these equations. Numerical experiments validate the theoretical findings, and the results indicate that computational speed can be improved by converting deep neural networks into shallow ones when solving differential equations.
Optimizing Thermal Lattice Boltzmann Method on an MT-3000 Processor with Neo-Heterogeneous Strategies
The Lattice Boltzmann Method (LBM) is a computational fluid dynamics method for simulating fluid flows with the benefits of locality and simplicity, making it ideal for parallel computing and complex flow simulations. This study focuses on developing a specialized Double Distribution Function (DDF) LBM software framework optimized for the MT-3000, a novel heterogeneous processor, to facilitate thermal incompressible flow simulations. To improve LBM’s performance on the complex multi-zone architecture of MT-3000, this paper introduces several innovative strategies. Firstly, a temporal fusion optimization strategy is implemented. This strategy involves postponing the temperature field calculations during time steps, efficiently decreasing the time overhead. Furthermore, we present “Pencil-H”, a novel pipelined algorithm meticulously designed to harness the unique capabilities of the MT-3000, thereby enhancing computational efficiency and communication effectiveness. Additionally, an architecture-aware multi-level parallelization algorithm is proposed, tailored to maximize the computational capabilities of the MT-3000. The effectiveness of these optimization strategies has been thoroughly validated through extensive bench-marking tests. These validations have shown remarkable performance enhancements, including a significant acceleration factor of 32.02X when compared to using 16 CPU cores. Notably, the optimized code demonstrated high-fidelity simulation capabilities for thermal incompressible flows, achieving 61.61% of the theoretical maximum performance defined by the roofline model.
A Conservative Sharp Interface Method for Compressible Two-Medium Flows on Unstructured Triangular Meshes
A conservative sharp interface method based on the cut-cell approach is proposed for the simulation of compressible two-medium flows on unstructured triangular meshes. In this method, the governing equations for each medium are solved using a MUSCL-type finite volume scheme. The exact two-medium Riemann problem is solved at the moving interface, while the local Lax–Friedrichs flux is employed along the fixed cell edges. The interface is tracked by solving the advection equation for the level set function, and explicit interface reconstruction is performed at each time step within the cut-cell framework. To address the small cell problem and ensure global conservation, a novel cell-merging algorithm, specifically designed for unstructured triangular meshes, is introduced to ensure that interface cells coincide with the interface. In contrast to existing cut-cell methods used on Cartesian meshes, the complexity of the new algorithm in constructing cut cells is significantly reduced. Additionally, the adaptive mesh refinement technology is employed to efficiently simulate complex flow problems while reducing the computational cost. A series of numerical tests is conducted to demonstrate the conservativeness and accuracy of the proposed method.
Should Exponential Integrators be Used for Advection-Dominated Problems?
In this paper, we consider the application of exponential integrators to problems that are advection dominated. In this context, we compare Leja and Krylov based methods to compute the action of exponential and related matrix functions. We set up a performance model by counting the different operations needed to implement the considered algorithms. This model assumes that the evaluation of the right-hand side is memory bound and allows us to evaluate performance in a hardware independent way. We find that exponential integrators, depending on the specific setting, either outperform or perform similarly to explicit Runge–Kutta schemes. We generally observe that Leja based methods outperform Krylov iterations in the problems considered. This is in particular true if computing inner products is expensive.
A New Composite Wall Function for Adverse Pressure Gradient Flows
This paper presents a composite wall function, which can be used for a wide range of the adverse pressure gradient (APG) and is derived from the composite walllaw provided in this study. The composite wall-law combines the characteristics of the log-law and the half-power law under the APG, and it degrades to the log-law under the zero pressure gradient (ZPG). A function relationship between the parameters in the composite wall-law and the dimensionless pressure gradient parameter $\mathcal{p}^+$ is provided based on a series of DNS data for the APG. Finally, a generalized wall function incorporating both the viscous sublayer and the buffer layer is presented. The new wall function can accurately predict the mean velocity of both equilibrium and nonequilibrium flows under the APG. The new composite wall function is validated using three test cases: NACA0012, 2D channel with a bump, and turbulent separation bubble. The results demonstrate the accuracy of the new composite wall function under the APG flows.
A Crank-Nicolson Finite Difference Scheme for the (2+1)D Saturable Nonlinear Schrödinger Equation with Generalized Damping
In this study, we implement a Crank-Nicolson finite difference scheme to discretize the (2+1)D saturable nonlinear Schrödinger equation with general damping. We show the existence and uniqueness of the discrete solution. The boundedness of the discrete mass and energy is established. The error between the exact and discrete solutions is shown to converge at a second-order rate in both time and space, according to the $L^2$ and $H^1$ discrete norms. Moreover, we show that the proposed scheme preserves the mass conservation and energy conservation for the (2+1)D saturable nonlinear Schrödinger equation without damping. Numerical simulations are conducted to validate these convergence properties and the conservation laws.
The Second-Order Mass-Conservative Characteristic Finite Difference Method for Sobolev Equation with Convection Term
In this work, a new Mass-Conservative Characteristic Finite Difference Method (MCC-FDM) is developed for solving the Sobolev equation. The new method combines the method of characteristics with mass-preserving interpolation, which not only keeps mass balance but also achieves second-order accuracy in both time and space. Numerical results are presented to confirm the temporal and spatial accuracy
A Discrete Collocation Technique using the Thin Plate Splines for Solving a Certain Class of Integro-Differential Equations Arising in Biology Models
The primary motivation of the current article is to study an approximate algorithm to solve a type of Volterra integro-differential equations involving nonlinearterms. These integro-differential equations have been utilized to simulate several realistic modeling arising in biological sciences, for example, the growth model of the toxins’ cumulative effects on a population residing in a closed system. Thin plate splines (TPSs), known as a subclass of radial basis functions (RBFs) independent of the shape parameter, create an efficient scheme to interpolate a function. Due to this desirable property, we use the discrete collocation approach with the TPS basis for estimating the solution of the mentioned integro-differential equations. Furthermore, the Gauss-Legendre integration formula is utilized to calculate the integrals emerging in the scheme. Remarkably, the algorithm of the offered approach can be comfortably performed on a PC with typical specifications derived from the simplicity of using TPSs. The researchers also investigate the convergence of the presented scheme. Some test examples including Volterra integro-differential equations are considered to make sure the veracity of the proposed scheme and the validity of the theoretical error analysis.
A Ciarlet-Raviart Mixed Finite Element Method for Optimal Control Problems Governed by a Fourth Order Bi-Wave Equation
The optimal control problem governed by a stationary fourth-order bi-wave equation is considered. To tackle this problem, we propose a bilinear mixed method of the Ciarlet-Raviart type. Our method exhibits an optimal convergence rate in the $L^2$-norm and demonstrates a global supercloseness property in the $H^1$ -seminorm. Moreover, through the application of an interpolation post-processing technique, the method achieves global superconvergence. We provide two numerical examples to numerically validate these theoretical properties of our proposed method.
On Stable Difference Schemes for the Solution of Soliton Type Coupled Sine-Gordon Equations in the Weak Sense
In the present paper numerical solution of the nonlinear coupled system of sine-Gordon equations is studied in the weak sense. A first-order accurate and two second-order accurate unconditionally stable difference schemes corresponding to the system of sine-Gordon equations are considered. Solutions of these difference schemes are presented in the space of distributions by variational methods. The fixed point theory and the finite difference method are combined in the numerical implementations, carried out in MATLAB, in order to verify the theoretical results.
ALM-PINN: An Adaptive Physical Informed Neural Network Optimized by Levenberg-Marquardt for Efficient Solution of Singular Perturbation Problems
This paper presents ALM-PINN, an adaptive physical informed neural network algorithm optimized by Levenberg-Marquardt. ALM-PINN is tailored to overcome challenges for solving singular perturbation problems (SPP). Traditional neural network-based solvers reframe solving differential equations task as a multi-objective optimization problem involving residual or Ritz error. However, significant disparities in the magnitudes of loss functions and their gradients frequency result in suboptimal training and convergence challenges. Addressing these issues, ALM-PINN introduces a learnable parameter for the perturbation parameter and constructs a two-terms loss function. The first loss term emphasizes approximating the governing equation, while the second term minimizes the difference between perturbation and learnable parameters. This adaptive learning strategy not only mitigates convergence issues in directly solving SPP but also alleviates the computational burden with asymptotic iteration from a large initial value. For one-dimensional tasks, ALM-PINN enhances training efficiency and reduces complexity by enforcing hard constraints on boundary conditions, streamlining the loss function sub-terms. The efficacy of ALM-PINN is evaluated on five SPPs, demonstrating its ability to capture sharp changes in physical quantities within the boundary layer, even with small perturbation coefficients. Furthermore, ALM-PINN exhibits reduced errors in both $L_ {\infty}$ and $L_2$ norms, coupled with improved convergence speed and stability.
Copyright (c) 2025 COPYRIGHT: © Global Science Press
On Convergence and Superconvergence of Discontinuous Galerkin Method for Semi-Explicit Index-1 Integro-Differential Algebraic Equations
This paper mainly focuses on the discontinuous Galerkin (DG) method for solving the semi-explicit index-1 integro-differential algebraic equation (IDAE), which is a coupled system of Volterra integro-differential equations (VIDEs) and second-kind Volterra integral equations (VIEs). The DG approach is applied to both the VIDE and VIE components of the system. The global convergence respectively in the $L^2$-norm and $L^\infty$-norm is established, and the local superconvergence for VIDE component is obtained. Furthermore, numerical examples are presented to validate the theoretical convergence and superconvergence results.
Integro-differential algebraic equations
Accurate Static and Dynamic Analysis Using Unstructured Meshes for Complicated Geometry Problems
The low accuracy using unstructured meshes to solve complicated geometry problems is a significant challenge in practical engineering. It is the key to solve this issue through enhancing the accuracy of tetrahedral element. This work develops a simple strategy to enhance the accuracy of tetrahedral element with the Shepard interpolation function. In which the strain field is reconstructed by introducing the weighted strains of adjacent tetrahedral elements to central tetrahedral element. The stiffness of the discrete system is significantly softened with this simple operation, which leads to a great accuracy improvement. Furthermore, this simple modification makes linear tetrahedral element owns higher accuracy even compared with hexahedral element. The high precision tetrahedral element makes finite element analysis more acceptable for engineers. It provides a novel solution to finite element analysis using unstructured meshes for practical engineering problems with complicated geometry. In order to validate the accuracy and feasibility of present modified tetrahedral element (M-T4) in complicated geometry problems, several engineering examples are performed, including linear static analysis, modal analysis, frequency response analysis, transient response analysis, and response spectrum analysis. The remarkable performance of the M-T4 suggests its wide applicability to practical engineering problems.
A Hybrid Reconstruction Method Based upon the Characteristic Fields Decomposition of Compressible Euler Equations
The local characteristic fields of compressible Euler equations can be decomposed into linearly degenerate and genuinely nonlinear, respectively. The former is associated with a contact discontinuity (the slip lines for multi-dimensional cases) and the latter is associated with shocks and rarefaction waves (Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 2009). The weighted essentially non-oscillatory (WENO) scheme can capture shock waves well but its resolution for short waves still needs to be improved, while one kind of new scheme constructed by the boundary variation diminishing (BVD) principle with the THINC (Tangent of Hyperbola for INterface Capturing) reconstruction can resolve the contact discontinuities with higher resolution but it may generate spurious numerical phenomena in some special cases. Combining the physical characteristics and the advantages of WENO and BVD schemes, this paper proposes a hybrid reconstruction method, in which the classical WENO scheme is used to discretize the genuinely nonlinear fields and the higher resolution BVD scheme is used to discretize the linearly degenerate field(s). Numerical experiments show that the hybrid method can preserve high accuracy near both contact discontinuities and short waves as the used higher resolution scheme while effectively avoiding spurious numerical phenomena (such as oscillations and overshoots). Compared with the BVD scheme, the hybrid method also improves computational efficiency, since it uses a more efficient WENO scheme in genuinely nonlinear fields.
Two Novel Space-Time Polynomial Particular Solutions Methods for the Time-Dependent Schrödinger Equation
The conventional method of polynomial particular solutions is only applicable to partial differential equations on the real number field. Building on this method, this paper proposes two novel approaches for numerical simulation of the time-dependent Schrödinger equation. Under the assumption of treating the time variable as an ordinary spatial variable, the first approach approximates the real and imaginary parts of the equation using two different sets of linear combinations, which are then substituted into the corresponding original governing equations to form a coupled differential equation. The second approach is derived based on the basic form of complex coefficient partial differential equations and involves polynomial particular solutions with imaginary terms. Using the same numerical examples, compared to conventional finite difference method, Fourier spectral method and radial basis function collocation method, this algorithm's stability is not limited by the grid ratio of the time step and spatial step. It is not only simple and feasible but also suitable for solving high-dimensional problems.
method of polynomial particular solutions.
A Fast Matrix Splitting Iteration Method for Fractional Regime-Switching Option Pricing Model
The discretization of the fractional regime-switching option pricing model leads to the linear system with the block diagonal and Kronecker product structure. A fast matrix splitting iteration method is presented to solve the discrete system. Theoretical analyses prove the convergence of the proposed iteration method. Numerical experiments show that the new method is efficient and outperforms the existing methods in terms of the number of iteration steps and the elapsed CPU time.
An Explicit Method for the Coupled Forward Backward Stochastic Differential Equations
This paper proposes an explicit method for the coupled forward backward stochastic differential equations (FBSDEs). Our method combines skillfully a weak second order stochastic Runge-Kutta method for solving forward equations with a two-step method for solving backward equations. We give a convergence theorem for the proposed method when the FBSDEs is weakly coupled (the forward equations are independent of the variable Z). Finally, some numerical results are presented, and the numerical results show that our method still works well even if the forward equations depend on Z.
Forward backward stochastic differential equations
Mechanisms for Stabilizing Thermal Lattice Boltzmann Equation and Their Applications for Convection-Diffusion Problem on Nonuniform Meshes
The thermal lattice Boltzmann equation (TLBE) has superior numerical stability as an explicit algorithm. However, its applications on nonuniform meshes are complicated. This paper clarifies the intrinsic mechanism for stabilizing computations in TLBE and proposes two solvers that combine the numerical stability of TLBE and flexible finite difference/volume schemes for nonuniform meshes. Through a brief review of the lattice Boltzmann method, it is concluded that the entropy increase of the collision operator is essential for numerical stability. This paper first proposes a macroscopic entropy-increasing (MEI) model for convection-diffusion problems by combining the MEI process and TLBE. The von Neumann stability analysis proves that the MEI model has no upper limit for mesh Fourier number. However, the accuracy of the MEI model is found to be sensitive to higher-order deviation terms. Therefore, a hybrid model that combines the MEI and the equilibrium-moment-based models is proposed to solve the problem. The von Neumann stability analysis demonstrates that the hybrid model can completely recover the numerical stability of TLBE. Numerical investigations validate the good stability and accuracy of the hybrid model. Most importantly, it can be easily applied to nonuniform meshes, whereas implementing TLBE on nonuniform meshes is relatively complicated.
Strong Converge Order of the General One-Step Method for Neutral Stochastic Delay Differential Equations under a Global Monotone Condition
We study the strong convergence of the general one-step method for neutral stochastic delay differential equations with a variable delay. First, we give the notions of C-stability and B-consistency, and then establish a fundamental theorem of strong convergence for the general one-step method solving the nonlinear neutral stochastic delay differential equations, where the corresponding diffusion coefficient with respect to the non-delay variables is highly nonlinear. Then, we construct the split-step backward Euler method which is a special implicit one-step method, and prove that it is C-stable, B-consistent, and strongly convergent of order 1/2. Finally, we give some numerical experiments to support the obtained results.
neutral stochastic delay differential equations.
An Absolutely Stabilized Virtual Element Method for the Incompressible Stokes Equations
In this paper, we propose an absolutely stabilized mixed virtual element method (mixed VEM) for the incompressible Stokes equations. We employ the $C^0$-conforming virtual element space for the velocity and discontinuous polynomial space for the pressure. The velocity-pressure pair contains "arbitrary-order" polynomials, and stability is guaranteed by the "absolutely stabilized finite element formulation" originally introduced in [21]. We establish error estimates in the energy norm for both velocity and pressure, as well as an $L^2$-error estimate for the velocity. Several numerical experiments validate the theoretical analysis.
Improved Sixth-Order WENO Finite Difference Schemes for Hyperbolic Conservation Laws
This article describes developing and improving sixth-order characteristic-wise Weighted Essentially Non-Oscillatory (WENO) finite difference schemes. These schemes are specially designed to solve scalar and system hyperbolic conservation laws with high accuracy/resolution and robustness. The schemes have been enhanced by using a new reference global smoothness indicator, which ensures the optimal order of accuracy for smooth solutions. The schemes also incorporate affine-invariant nonlinear Ai-weights that are independent of the scaling of solution and the choice of sensitivity parameter. The improved nonlinear weights enhance the essentially non-oscillatory (ENO) capturing of discontinuities and minimize the numerical dissipation, especially for long-time simulations. The study also introduces the positivity-preserving limiter to ensure that the numerical solution of Euler equations is physically valid. The effectiveness of improved schemes is demonstrated through one- and two-dimensional benchmark shock-tube problems, such as the Sod, Lax, and Woodward-Colella problems. The improved schemes are compared with other WENO schemes in terms of accuracy, resolution, ENO, and robustness.
Adaptive Finite Element Solutions of a Diffusion Model for Drug Release in PLGA Microspheres
Poly (lactic-co-glycolic acid) copolymer (PLGA) microspheres are crucial in drug delivery due to their ability to provide controlled release of a wide range of therapeutic agents while minimizing side effects and enhancing biocompatibility. In this paper, a mathematical model is proposed and numerically studied for the drug release process in PLGA microspheres. A three-dimensional diffusion model is constructed based on Fick’s law, being different from the traditional one-dimensional model in the past and fitting the configuration from the drug delivery system, while an h-adaptive finite element method is designed and realized using a C++ library AFEPack for numerical solutions of the proposed model. A notable feature is that the h-adaptivity makes the proposed model and numerical method a quality tool for studying concerned phenomena such as initial bursts in drug delivery. In numerical experiments, besides demonstrating the convergence of the numerical method, a case on progesterone release from PLGA is also studied using the proposed model, from which the potential of the model and the method towards the practical application is shown successfully.
Weighted Sonine Conditions and Application
The Sonine kernel described by the classical Sonine condition of convolution form is an important class of kernels used in integral equations and nonlocal differential equations. This work extends this idea to introduce weighted Sonine conditions where the non-convolutional weight functions accommodate the inhomogeneity in practical applications. We characterize tight relations between classical Sonine condition and its weighted versions, which indicates that the non-degenerate weight functions may not introduce significant changes on the set of Sonine kernels. To demonstrate the application of weighted Sonine conditions, we employ them to derive equivalent but more feasible formulations of weighted integral equations and nonlocal differential equations to prove their well-posedness, and discuss possible application to corresponding partial differential equation models.
Taylor Remainder-Based Discontinuity Detection Method with Application to Hybrid Compact-WENO Finite Difference Scheme for Hyperbolic Conservation Laws
In this study, the Taylor Remainder-based Discontinuity Detection Method (TRDDM) is designed to identify shocks in numerical solutions of hyperbolic conservation laws. The TRDDM leverages the fact that Taylor remainder is minimal in smooth regions but significantly large at discontinuities. This method reduces the need for threshold parameters, making it less complex. Pre-processing the target dataset involves normalization, detection of constants, and identification of high-frequency waves. After pre-processing, Taylor remainder is calculated for the remaining dataset that contains discontinuous patterns. An appropriate partition point separates the error components, with larger errors indicating discontinuities. TRDDM analyzes local features of numerical solutions and is capable of capturing even minor discontinuities. The accuracy and efficiency of the hybrid compact-WENO scheme with TRDDM are demonstrated through classical one- and two-dimensional shocked problems.
Two Lattice Boltzmann Models for Three-Dimensional Coupled Burgers’ Equations
Two lattice Boltzmann models are presented for the general three-dimensional coupled Burgers’ equations with different diffusion coefficients. The coupled parts or all of the convection items are treated as the source terms, where the spatial gradient can be calculated by the distribution function. The three-dimensional coupled Burgers’ equations can be exactly recovered from the lattice Boltzmann(LB) models through the direct Taylor expansion. Some numerical tests were performed to verify the accuracy and efficiency of the present models.
three-dimensional coupled Burgers’ equations
Fluid-Structure Interactions: One-Field Monolithic Fictitious Domain Method and Its Parallelization
In this paper, we delve into the parallelization of the one-field monolithic fictitious domain (MFD) method, a comprehensive algorithm designed for simulating fluid-structure interactions (FSI). This method treats the velocity fields of both the fluid and solid as a unified field through an $L^2$ projection, enabling synchronous equation solving instead of sequential approaches. Given the computational intensity of 3D domain simulations, parallelizing the algorithm and the linear systems derived from discretization becomes imperative. This parallelization significantly reduces simulation time from several months to just a few days. We focus on parallelizing the algorithm for FSI simulations on uniform meshes. Our primary aim is to develop an efficient and robust algorithm capable of handling complex FSIs, including scenarios involving collisions of multiple immersed solids in fluid. Such scenarios often require high-resolution meshes to accurately resolve phenomena near collision points and fluid-structure interfaces.