Unconditional Stability Research Articles

Unconditionally energy stable high-order BDF schemes for the molecular beam epitaxial model without slope selection

In this paper, we consider a class of k-order (3≤k≤5) backward differentiation formulas (BDF-k) for the molecular beam epitaxial (MBE) model without slope selection. Convex splitting technique along with k-th order Douglas-Dupont regularization term τnk(−Δ)kD_kϕn (D_k represents a truncated BDF-k formula) is added to the numerical schemes to ensure unconditional energy stability. The stabilized convex splitting BDF-k (3≤k≤5) methods are unique solvable unconditionally. Then the modified discrete energy dissipation laws are established by using the discrete gradient structures of BDF-k (3≤k≤5) formulas and processing k-th order explicit extrapolations of the concave term. In addition, based on the discrete energy technique, the L2 norm stability and convergence of the stabilized BDF-k (3≤k≤5) schemes are obtained by means of the discrete orthogonal convolution kernels and the convolution type Young inequalities. Numerical results are carried out to verify our theory and illustrate the validity of the proposed schemes.

Numerical study of the multi-dimensional Galilei invariant fractional advection–diffusion equation using direct mesh-less local Petrov–Galerkin method

This article presents a local mesh-less procedure for simulating the Galilei invariant fractional advection–diffusion (GI-FAD) equations in one, two, and three-dimensional spaces. The proposed method combines a second-order Crank–Nicolson scheme for time discretization and the second-order weighted and shifted Grünwald difference (WSGD) formula. This time discretization scheme ensures unconditional stability and convergence with an order of O(τ2). In the spatial domain, a mesh-less weak form is employed based on the direct mesh-less local Petrov–Galerkin (DMLPG) method. The DMLPG method employs the generalized moving least-square (GMLS) approximation in conjunction with the local weak form of the equation. By utilizing simple polynomials as shape functions in the GMLS approximation, the necessity for complex shape function construction in the MLS approximation is eliminated. To validate and demonstrate the effectiveness of the proposed algorithm, a variety of problems in one, two, and three dimensions are investigated on both regular and irregular computational domains. The numerical results obtained from these investigations confirm the accuracy and reliability of the developed approach in simulating GI-FAD equations.

A High-Order Numerical Method Based on a Spatial Compact Exponential Scheme for Solving the Time-Fractional Black–Scholes Model

This paper investigates a high-order numerical method based on a spatial compact exponential scheme for solving the time-fractional Black–Scholes model. Firstly, the original time-fractional Black–Scholes model is converted into an equivalent time-fractional advection–diffusion reaction model by means of a variable transformation technique. Secondly, a novel high-order numerical method is constructed with (2−α) accuracy in time and fourth-order accuracy in space based on a spatial compact exponential scheme, where α is the fractional derivative. The uniqueness of solvability of the derived numerical method is rigorously discussed. Thirdly, the unconditional stability and convergence of the derived numerical method are rigorously analyzed using the Fourier analysis technique. Finally, numerical examples are presented to test the effectiveness of the derived numerical method. The proposed numerical method is also applied to solve the time-fractional Black–Scholes model, whose exact analytical solution is unknown; numerical results are illustrated graphically.

An unconditionally stable scheme for the immersed boundary method with application in cardiac mechanics

The stability of the immersed boundary (IB) method is a challenge in simulating fluid–structure interaction problems, where time step constraints are significantly stricter than in pure fluid simulations. We propose a novel unconditionally stable scheme for the immersed boundary finite element (IBFE) method. The structure is handled implicitly and characterized by strain energy functions, rather than being modeled as fibers or membranes. Through energy estimate, we prove the unconditional stability of the fully discrete approximation in the absence of the convective term. In real simulations of cardiac mechanics problems, the time step is much larger, only limited by the Courant–Friedrichs–Lewy condition of the fluid. The novelty of this work lies in the combination of dual interpolation and distribution operators, the Jacobian-free Newton–Krylov method for solving nonlinear algebraic systems, and the semi-Lagrangian method for handling the convective term. To validate the effectiveness and accuracy of our approach, we present various benchmarks and conduct a quasi-static simulation of a three-dimensional real left ventricular model. We have shown that the numerical stability of our scheme is very robust even with much larger time step compared to conventional explicit IB methods. Our work paves the way for further works on efficient solvers of the IBFE method.

Discussing the limitations of unconditional stability indicators in evaluating thermoacoustic quality of burners with flames

Assessing the thermoacoustic performance of designed combustors, with a focus on the stability quality factor, is crucial. Thermoacoustic instability in combustion appliances arises from intricate interactions among unsteady combustion, heat transfer, and (maybe) acoustic modes within the system. Accurate prediction of system stability requires modeling all components, including the burner with flame. Traditionally, the burner in the presence of combustion is represented as an acoustically (active) two-port block with passive upstream and downstream acoustic terminations. The dispersion relation of the thermoacoustic system is commonly used for anticipating eigen-frequencies and assessing stability. However, practical scenarios often lack specific information about upstream and downstream terminations during development. This raises a critical question: How can the thermoacoustic performance of burners and their associated flames be evaluated without specified acoustics? This article addresses this question by exploring the concept of unconditional stability in a generic two-port thermoacoustic system. The unconditional stability criteria have been used as quality indicators in designing electrical devices. This rich toolbox has been introduced in thermoacoustics. We first scrutinize assumptions underlying two most known unconditional stability-based criteria called [Formula: see text] and [Formula: see text] factors, connecting them to the general thermoacoustic problems. Then, the application of these criteria in assessing the thermoacoustic quality of burners with flames are discussed. This investigation revealed that while they are able to accurately predict the histogram of unstable frequencies and critical frequency bands, their use as reliable indicators to assess thermoacoustic quality in burners are not recommended due to their mathematical limitations and high level of conservatism of these factors.

A dimension reduction method of two‐grid finite element solution coefficient vectors for the Allen–Cahn equation

This paper mainly focuses on the dimension reduction of unknown finite element (FE) solution coefficient vectors in two‐grid Crank–Nicolson FE (TGCNFE) method for the nonlinear Allen–Cahn equation. For this purpose, a new TGCNFE method for the nonlinear Allen–Cahn equation is first developed and the unconditional stability and errors of TGCNFE solutions are analyzed. Thereafter, the most important thing is to lower the dimension of the unknown FE solution coefficient vectors by a proper orthogonal decomposition, to establish a new reduced‐dimension extrapolated TGCNFE (RDETGCNFE) method, and to analyze the unconditional stability and errors of RDETGCNFE solutions. Moreover, the correctness of our theoretical results is validated by some numerical experiments.

Vertex Block Descent

We introduce vertex block descent, a block coordinate descent solution for the variational form of implicit Euler through vertex-level Gauss-Seidel iterations. It operates with local vertex position updates that achieve reductions in global variational energy with maximized parallelism. This forms a physics solver that can achieve numerical convergence with unconditional stability and exceptional computation performance. It can also fit in a given computation budget by simply limiting the iteration count while maintaining its stability and superior convergence rate. We present and evaluate our method in the context of elastic body dynamics, providing details of all essential components and showing that it outperforms alternative techniques. In addition, we discuss and show examples of how our method can be used for other simulation systems, including particle-based simulations and rigid bodies.

Asymptotic freedom in the lattice Boltzmann theory.

Asymptotic freedom is a feature of quantum chromodynamics that guarantees its well posedness. We derive an analog of asymptotic freedom enabling unconditional linear stability of lattice Boltzmann simulation of hydrodynamics. We further demonstrate the validity of the derived conditions via the special case of the equilibrium based on entropy maximization, which is shown to be uniquely renormalizable.

A second-order in time, BGN-based parametric finite element method for geometric flows of curves

Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett et al. (2007) [8], due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which integrates with two different mesh regularization techniques, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as mesh regularization techniques, our proposed second-order schemes exhibit good properties with respect to the mesh distribution. In addition, an unconditional interlaced energy stability property is obtained for one of the mesh regularization techniques.

On the stability of θ-methods for DDEs and PDDEs

In this paper, the stability of θ-methods for delay differential equations is studied based on the test equation y′(t)=−Ay(t)+By(t−τ), where τ is a constant delay and A is a positive definite matrix. It is mainly considered the case where the matrices A and B are not simultaneosly diagonalizable and the concept of field of values is used to prove a sufficient condition for unconditional stability of these methods and another condition which also guarantees their stability, but according to the step size. The results obtained are also simplified for the case where the matrices A and B are simultaneously diagonalizable and compared with other similar works for the general case. Several numerical examples in which the theory discussed here is applied to parabolic problems given by partial delay differential equations with a diffusion term and a delayed term are presented, too.

The high-order exponential semi-implicit scalar auxiliary variable approach for the general nonlocal Cahn-Hilliard equation

The nonlocal Cahn-Hilliard equation with nonlocal diffusion operator is more suitable for the simulation of microstructure phase transition than the local Cahn-Hilliard equation. In this paper, based on the exponential semi-implicit scalar auxiliary variable method, the highly efficient and accurate schemes (in time) with unconditional energy stability for solving the nonlocal Cahn-Hilliard equation are proposed. On the one hand, we have demonstrated the unconditional energy stability and mass conservation for the nonlocal Cahn-Hilliard equation with its high-order numerical schemes in the continuous and discrete level carefully and rigorously. On the other hand, in order to reduce the calculation and storage cost in numerical simulation, we use the fast solver based on fast Fourier transform and conjugate gradient approach for spatial discretization. Some numerical simulations involving the Gaussian kernel are presented and show the stability, accuracy, efficiency and unconditional energy stability of the proposed schemes.

Decoupled and unconditionally stable iteration method for stationary Navier–Stokes equations

AbstractIt is well known the Oseen iteration for the stationary Navier–Stokes equations is unconditionally stable. However, it is a coupled type scheme where the velocity and pressure are coupled together at each iteration. By treating pressure explicitly would lead to a decoupled iteration, but this treatment is unstable. In this article, we construct a decoupled and unconditionally stable iteration method to solve the stationary Navier–Stokes equations by adopting the pressure projection method to the temporal disturbed Navier–Stokes system whose solution approximates the steady state solution over time (). We also rigorously prove its unconditional stability. Numerical simulations demonstrate that our iterative method is more efficient and stable than the extensively used T‐S and Oseen iterations, and could solve the fluid flow with high Reynolds number.

A Modified Implicit Monte Carlo Method for Frequency-Dependent Three Temperature Radiative Transfer Equations

In this work, we develop a modified implicit Monte Carlo method for the frequency-dependent (multigroup) three-temperature (3T) radiative transfer equations. A new reformulation of the 3T model is utilized to deal with the coupling between the electron and ion energies. This will result in an effective scattering term, together with a combination source term of electron and ion energies, which can be resolved by the Monte Carlo simulations. The method is proven to retain the asymptotic preserving property in the limit where the electron-ion coupling coefficient tends to zero or infinity. Furthermore, numerical simulations confirm the unconditional stability of the new method as well as its ability to preserve the total energy conservation. Several numerical results are presented to showcase the efficacy of this new method.

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

Herein, we mainly develop a new reduced-dimension method for the unknown finite element (FE) solution coefficient vectors to the elastic wave equation containing the known singular source term, the unknown displacement vector, and the unknown stress tensor matrix. For this end, we need first to develop the Crank-Nicolson (CN) FE (CNFE) method with unconditional stability, unconditional convergence, and second-order time accuracy for the elastic wave equation and analyze the existence, unconditional stability, and errors of the CNFE solutions. After that we employ a proper orthogonal decomposition to lower the dimensionality of the unknown FE solution coefficient vectors in the CNFE method for the elastic wave equation and to design a new reduced-dimension extrapolated CNFE (RDECNFE) method. Subsequently, we employ matrix analysis to discuss the existence, unconditional stability, unconditional convergence, and errors of the RDECNFE solutions. Finally, we use two numerical examples to show the superiority of the RDECNFE method and our theoretical results. It is also shown that the RDECNFE method is feasible and effective for solving the elastic wave equation.

Stability of hybrid time integration scheme for Lord–Shulman thermopiezoelectricity

Based on the existing initial boundary value and variational problems of Lord–Shulman thermopiezoelectricity a transient analysis of the behavior of piezoelectric materials is performed numerically. For space discretization of the variational problem the finite element method is used and for time discretization a hybrid time integration scheme is constructed on the basis of Newmark scheme for hyperbolic equation and generalized trapezoidal rule for parabolic equations. The unconditional stability of the developed time integration scheme is proved for some specific values of the scheme parameters by making use of energy balance law for the obtained time-discretized variational problem. Finally, the applicability of the constructed numerical scheme is demonstrated by the results of the numerical experiment which are compared with the ones available in literature.

Asymptotic preserving semi-Lagrangian discontinuous Galerkin methods for multiscale kinetic transport equations

We propose a new family of asymptotic preserving (AP) discontinuous Galerkin (DG) schemes for kinetic transport equations under a diffusive scaling. They are built upon a characteristics-based model reformulation proposed in Zhang et al. (2023) [27]. The reformulated model comprises a convection-diffusion-like equation for the macroscopic density and another evolution equation for the distribution function. We employ the equations in a weak form to enjoy the advantages of a DG method. We also leverage a semi-Lagrangian (SL) approach to achieve higher efficiency than a fully implicit scheme without loss of unconditional stability. A damping technique is introduced to control spurious numerical oscillations for high-order DG methods. We establish formal AP and Fourier stability analyses for fully discrete schemes in a linear scenario. Numerical experiments, covering one-dimensional to three-dimensional in space problems, confirm the accuracy, AP property, uniformly unconditional stability, and high parallel efficiency of the proposed methods.

Time two-grid fitted scheme for the nonlinear time fractional Schrödinger equation with nonsmooth solutions

In this paper, we delve into the regularity properties and the development of an efficient numerical strategy for addressing the nonlinear time-fractional Schrödinger equation. Initially, we embark on an examination of the solution’s regularity, followed by an investigation into regularity enhancement through the application of the Laplace and finite Fourier sine transforms. Subsequently, after the decomposition of u-u0, we introduce an innovative time two-grid fitted scheme designed for the equation’s resolution, which demonstrates unconditional stability and convergence in the L2 norm, achieving the global convergence order of O(τF2+τC4+h2). Here, τF and τC denote the temporal step sizes on the fine and coarse grids, respectively, while h represents the spatial step size. Notably, this is the inaugural application of such an approach for solving the nonlinear time fractional Schrödinger equation. Ultimately, our numerical experiments validate that this method not only retains computational efficiency but also significantly enhances convergence accuracy.

A New Reduced-Dimension Iteration Two-Grid Crank–Nicolson Finite-Element Method for Unsaturated Soil Water Flow Problem

The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design a new reduced-dimension iteration TGCNFE (RDITGCNFE). For this objective, a new time semi-discrete CN (TSDCN) scheme for the nonlinear unsaturated soil water flow problem is first designed and the existence, stability, and error estimates of TSDCN solutions are demonstrated. Subsequently, a new TGCNFE format for the nonlinear unsaturated soil water flow problem is designed and the existence, unconditional stability, and error estimates of TGCNFE solutions are demonstrated. Next, a new RDITGCNFE format with the same FE basis functions as the TGCNFE format is built by the POD method and the existence, unconditional stability, and error estimates of RDITGCNFE solutions are discussed. Ultimately, the rightness of theory results and the superiority of the RDITGCNFE format are verified by two sets of numerical tests. It is worth noting that the RDITGCNFE format differs completely from all previous reduced-dimension methods, including the authors’ previous works. Therefore, the study of this paper can not only provide a new theoretical method for the dimensionality reduction of numerical models for nonlinear problems but also provide an algorithm implementation technology for the numerical simulation of practical engineering problems.

Unconditional Stability and Fourth-Order Convergence of a Two-Step Time Split Explicit/ Implicit Scheme for Two-Dimensional Nonlinear Unsteady Convection-Diffusion-Reaction Equation

Unconditional Stability and Fourth-Order Convergence of a Two-Step Time Split Explicit/ Implicit Scheme for Two-Dimensional Nonlinear Unsteady Convection-Diffusion-Reaction Equation

A high-order upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier–Stokes equations

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.