Time Step Size Research Articles

Learning Reduced Fluid Dynamics

Predicting the state evolution of ultra high-dimensional, time-reversible fluid dynamic systems is a crucial but computationally expensive task. Existing physics-informed neural networks either incur high inference cost or cannot preserve the time-reversible nature of the underlying dynamics system. We propose a model-based approach to identify low-dimensional, time reversible, nonlinear fluid dynamic systems. Our method utilizes the symplectic structure of reduced Eulerian fluid and use stochastic Riemann optimization to obtain a low-dimensional bases that minimize the expected trajectory-wise dimension-reduction error over a given distribution of initial conditions. We show that such minimization is well-defined since the reduced trajectories are differentiable with respect to the subspace bases over the entire Grassmannian manifold, under proper choices of timestep sizes and numerical integrators. Finally, we propose a loss function measuring the trajectory-wise discrepancy between the original and reduced models. By tensor precomputation, we show that gradient information of such loss function can be evaluated efficiently over a long trajectory without time-integrating the high-dimensional dynamic system. Through evaluations on a row of simulation benchmarks, we show that our method reduces the discrepancy by 50-90 percent over conventional reduced models and we outperform PINNs by exactly preserving the time reversibility.

Effect of Different Time Step Sizes on Pedestrian Evacuation Time under Emergencies Such as Fires Using an Extended Cellular Automata Model

The cellular automata (CA) model has been a meaningful way to study pedestrian evacuation during emergencies, such as fires, for many years. Although the time step used in the CA model is one of the most essential elements, there is a lack of research on its impact on evacuation time. In this paper, we set different time step sizes in an extended cellular automaton model and discuss the effect of time step size on the overall evacuation time under different emergency types and levels. For a fixed step time mode, the larger the time step, the longer the evacuation time. In each time step size, the evacuation time gradually increases with the increase of emergency level, and there is a sharp increase when the time for pedestrians to move one step is exactly an integer multiple of the time step. When there is no friction between pedestrians, the evacuation time at each time step first decreases slightly with the increase of emergency level and then remains unchanged; the larger the time step, when the evacuation time remains unchanged, the lower the emergency level and the greater the evacuation time. For the variable time step model, when the friction between pedestrians approaches infinity, the total evacuation time does not change with the emergency level; when the friction between pedestrians is reduced, the total evacuation time slightly decreases with the increase of the emergency level. The less friction there is, the more significant the reduction. The results of previous actual experiments are also reflected in the simulation at a lower emergency level. The result shows that the time step size significantly impacts the evacuation simulation results of the CA model, and researchers should choose carefully to obtain more realistic simulation results.

Implicit-explicit time integration for the immersed wave equation

Efficient solution strategies for wave propagation problems with complex geometries are essential in many engineering fields. Immersed boundary methods simplify mesh generation by embedding the domain of interest into an extended domain that is easy to mesh, introducing the challenge of dealing with cells that intersect the domain boundary. We consider the finite cell method that extends the weak form from the physical into the extended domain, multiplied by a small value, to stabilize badly cut cells. Combined with explicit time integration methods, the presence of finite cells with very little support in the physical domain results in tiny critical time step sizes. While the finite cell stabilization limits how small the critical time step size can become, this limit is still restrictive for many applications. Explicit transient analyses commonly use the spectral element method due to its natural way of obtaining diagonal mass matrices through nodal lumping. The resulting method is very efficient since nodal lumping renders the solution of the equation systems trivial while not reducing the accuracy or the critical time step size. The combination of the spectral element and finite cell methods is called the spectral cell method. Unfortunately, a direct application of nodal lumping in the spectral cell method is impossible due to the special quadrature necessary to treat the discontinuous integrand inside the cut cells. The existing approaches to lump the mass matrices of cut cells significantly reduce the accuracy of the approximation.We analyze an implicit-explicit (IMEX) time integration method to exploit the advantages of the nodal lumping scheme for uncut cells on one side and the unconditional stability of implicit time integration schemes for cut cells on the other. In this hybrid, immersed Newmark IMEX approach, we use explicit second-order central differences to integrate the uncut degrees of freedom that lead to a diagonal block in the mass matrix and an implicit trapezoidal Newmark method to integrate the remaining degrees of freedom (those supported by at least one cut cell). The immersed Newmark IMEX approach preserves the high-order convergence rates and the geometric flexibility of the finite cell method while retaining the efficiency of the nodal lumping scheme for uncut cells without compromising the critical time step size. We analyze a simple system of spring-coupled masses to highlight some of the essential characteristics of Newmark IMEX time integration. We then solve the scalar wave equation with immersed boundaries on two- and three-dimensional examples with significant geometric complexity to show that our approach is more efficient than state-of-the-art time integration schemes when comparing accuracy and runtime. While we focus on the finite and spectral cell methods, we expect other immersed methods to benefit equally from Newmark IMEX time integration.

A time two-grid algorithm for two-dimensional nonlinear time-fractional partial integro-differential equations

In this paper, a temporal second order two-grid difference scheme is proposed for the two-dimensional nonlinear time-fractional partial integro-differential equations with a weakly singular kernel. The first-order backward difference and L1 formula are used in the temporal direction to estimate the first level of time, the L2−1σ formula and L1-type formula are used in the temporal direction for later time steps, and the central difference formula is used in the spatial directions. To improve the computational efficiency of nonlinear system, an efficient time two-grid algorithm is proposed. This algorithm firstly solves a nonlinear system on the coarse grid, and then the Lagrangian linear interpolation is applied on the coarse grid to estimate the function values on the fine grid. The stability and convergence of the two-grid difference scheme are analyzed by the energy method. The convergence order of the two-grid difference scheme is O(τF2+τC4+hx2+hy2), where τF and τC are the time step sizes of fine grid and coarse grid respectively, while hx and hy are the space step sizes. Numerical experiments show that the accuracy of the theoretical analysis and the efficiency of the two-grid algorithm.

Unconditional optimal first‐order error estimates of a full pressure segregation scheme for the magnetohydrodynamics equations

AbstractIn this article, a first‐order linear fully discrete pressure segregation scheme is studied for the time‐dependent incompressible magnetohydrodynamics (MHD) equations in three‐dimensional bounded domain. Based on an incremental pressure projection method, this scheme allows us to decouple the MHD system into two sub‐problems at each time step, one is the velocity‐magnetic field system, the other is the pressure system. Firstly, a coupled linear elliptic system is solved for the velocity and the magnetic field. Next, a Poisson‐Neumann problem is treated for the pressure. We analyze the temporal error and the spatial error, respectively, and derive the temporal‐spatial error estimates of for the velocity and the magnetic field in the discrete space and for the pressure in the discrete space without imposing constraints on the mesh width and the time step size . Finally, some numerical results are presented to confirm the theoretical predictions and demonstrate the efficiency of the method.

Fast explicit time integration schemes for parabolic problems in mechanics

AbstractWe present a family of fast explicit time integration schemes of first, second and third order accuracy for parabolic problems in mechanics solved via standard numerical methods that have considerable higher computational efficiency versus existing explicit methods of the same order. The derivation of the new explicit schemes is inspired on the finite increment calculus (FIC) procedure used for obtaining stabilized numerical schemes in fluid and solid mechanics. The new (so‐called) explicit FIC‐Time (EFT) schemes allow considerable larger time steps than the standard first order forward Euler (FE) scheme and the second and third order Adams–Bashforth schemes. The comparison with Runge–Kutta schemes also favors the FIC‐Time schemes in terms of the limit time step size (for second order schemes) and the total number of matrix‐vector multiplications per time step (for second and third order schemes). The new first order explicit schemes have a faster convergence to steady‐state than the FE scheme. The accuracy and efficiency of the new EFT schemes are verified in examples of application to the transient heat conduction equation using the finite element method.

Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations

Abstract We solve two numerical experiments described by 2D nonconstant coefficient advection–diffusion equations with specified initial and boundary conditions. Three finite difference methods, namely Lax–Wendroff, Du-Fort–Frankel and a nonstandard finite difference scheme, are derived and used to solve the two problems, whereby only the first problem has an exact solution. Stability analysis is performed to obtain a range of values of the time step size at a fixed spatial step size. We obtain the rate of convergence in space when the three methods are used to solve Problem 1. Computational times of the three algorithms are computed for Problem 1. Results are displayed for the two problems using the three methods at times T = 1.0 T=1.0 and T = 5.0 T=5.0 . The main novelty is the stability analysis, which is not straightforward as we are working with numerical methods discretising 2D nonconstant coefficient advection–diffusion equation where many parameters are involved. The second highlight is to determine the most efficient scheme from the three methods. Third, there are very few published studies on analysis and use of numerical methods to solve nonconstant coefficient advection–diffusion equations, and this is one of the very few rare articles treating such topics.

A new 2D ESPH bedload sediment transport model for rapidly varied flows over mobile beds

A 2D Eulerian meshless bedload sediment transport model is developed using smoothed particle hydrodynamics (SPH) to simulate rapidly varied flows over mobile beds. In the developed model, we adopt a weakly coupled numerical approach to explicitly solve the governing equations, including 2D shallow water equations for fluid flow motion and the Exner equation for bed sediment movement at the same time step. A defined virtual bedload velocity in a weakly coupled approach is involved in the calculation of time step sizes. However, nonphysical virtual bedload velocities, which often occur in cases with nearly flat beds, require extremely small time step sizes. A formulation of the lower and upper bounds of information propagation speeds for an HLLC approximate Riemann solver is thus proposed to remedy the problem. Four case studies involving dam break flows in prismatic and erodible channels, knickpoint migration, dam erosion due to overtopping flow and dam-break flow in an erodible channel with an abrupt expansion are adopted to validate the developed model. In addition, to compare the performances of different particle interaction configurations under Cartesian uniform particle arrangements, four and eight interacting particles are obtained by changing the smoothing length. Against the measured results, the case of eight interacting particles shows more accurate predictions of erosion peaks along the measured cross-sections because the lateral flows are significant. The good agreement between the simulated and measured results shows that the developed 2D Eulerian SPH bedload sediment transport model with eight interacting particles is well suited for simulating complicated flow-induced bed erosion.

Shape transformation on curved surfaces using a phase-field model

Shape transformation on evolving curved surfaces is essential for its diverse applications across various scientific disciplines and facilitates the deeper understanding of natural phenomena, the development of new materials, and engineering design optimization. In this study, we develop a phase-field model and its numerical methods for shape transformation on curved surfaces. A modified surface Allen–Cahn (AC) equation with a fidelity term is proposed to simulate shape transformation on curved surfaces. To numerically solve the modified surface AC equation on curved surfaces, we propose a fully explicit scheme and an unconditionally stable method. The proposed stable approach is not only simple and efficient to implement numerically but is also unconditionally stable and eliminates the restrictive temporal time step size constraints. Through numerical experiments using the proposed approach, we demonstrate that shape transformation on evolving curved surfaces can be implemented on both simple and complex curved surfaces.

A unified stochastic particle method with spatiotemporal adaptation for simulating multiscale gas flows

Recently, the Unified Stochastic Particle (USP) method has emerged as a promising approach for multiscale particle simulations, tailored specifically for non-equilibrium gas flow scenarios. However, in typical simulations of multiscale flow fields, the time step and grid size of the original USP method remain constrained by the smallest spatiotemporal scale of local flow field, limiting its true multiscale simulation potential. To overcome this critical issue, this work introduces a spatiotemporal adaptive USP method. Initially, we analyze the sources of errors in the traditional single-scale stochastic particle methods under large time steps, which leads to the derivation of the governing equations of USP. Building upon this foundation, we propose principles that dictate the rational time step and grid size for an arbitrary USP simulation. Subsequently, we introduce a spatiotemporal adaptive approach primarily based on the local gradient length, leading to the development of a USP solver endowed with spatiotemporal adaptability. The universality, reliability, and efficiency of the algorithm have been verified through an examination of three cases: two-dimensional flow over a cylinder, three-dimensional flow over a blunted body, and flow over a scaled X38-like model. This work forms an integral component of the development initiative for the general USP solver—SPARTACUS—with relevant source code being open-source in an online repository under the GNU General Public License (GPL).

Conjugate gradient method with regularization in estimating mold surface heat flux during continuous casting

The reconstruction of the boundary heat flux between solidifying steel and water-cooled mold wall using measured temperature data is recognized as an inverse heat conduction problem (IHCP). A Tikhonov spatial regularization approach to enhance the smoothness of the Conjugate Gradient Method (CGMr) was proposed which incorporates three different orders of spatial regularization: zeroth, first, and second order. The optimal regularization parameter was selected using a modified L-curve method. The accuracy of the CGMr is investigated with Case 1: a triangular time-spatial variation heat flux, and Case 2: a step change in the form of rectangular variation heat flux. The effects of measurement noise, grid points and time-step size are investigated. The results show that the minimum relative error (eRMS) of the predicted Case 1 heat flux is 1.98%, 7.64%, and 8.02% for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 2 heat flux are 3.82%, 13.42%, and 14.91%. Subsequently, CGMr algorithm is applied to calculate the heat flux in a mold simulator experiment. By comparing the relationship between heat fluxes reconstructed by CGMr after different iteration numbers, it is observed that the recovered heat flux of 2.14 MW/m2 with zeroth regularization remains highly stable.

Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral-Element Methods

We extend the monolithic convex limiting (MCL) methodology to nodal discontinuous Galerkin spectral-element methods (DGSEMS). The use of Legendre-Gauss-Lobatto (LGL) quadrature endows collocated DGSEM space discretizations of nonlinear hyperbolic problems with properties that greatly simplify the design of invariant domain-preserving high-resolution schemes. Compared to many other continuous and discontinuous Galerkin method variants, a particular advantage of the LGL spectral operator is the availability of a natural decomposition into a compatible subcell flux discretization. Representing a high-order spatial semi-discretization in terms of intermediate states, we perform flux limiting in a manner that keeps these states and the results of Runge-Kutta stages in convex invariant domains. In addition, local bounds may be imposed on scalar quantities of interest. In contrast to limiting approaches based on predictor-corrector algorithms, our MCL procedure for LGL-DGSEM yields nonlinear flux approximations that are independent of the time-step size and can be further modified to enforce entropy stability. To demonstrate the robustness of MCL/DGSEM schemes for the compressible Euler equations, we run simulations for challenging setups featuring strong shocks, steep density gradients, and vortex dominated flows.

Challenges in Numerical Solutions of Higher-Dimensional Differential Equations

Differential equations constitute a fundamental tool in modeling various natural phenomena across scientific disciplines such as physics, engineering, and finance. We provide an overview of fractional differential equations, focusing on the computational requirements associated with their numerical solutions from a computer science perspective. We analyze the computational intricacies concerning First-Order Linear ODE, First-Order Nonlinear ODE, Second-Order Linear ODE, Second-Order Nonlinear ODE, Heat Equation (PDE), and Wave Equation (PDE). This comparative assessment delves into the computational demands of solving these equations using differential equation methodologies. While analytical solutions provide deep insights, obtaining numerical solutions, particularly in higher dimensions, remains a persistent challenge. Finite difference methods commonly employed for numerical solutions, In higher-dimensional problems, traditional numerical methods face challenges stemming from an exponential surge in grid points and the consequent demand for substantially decreased time step sizes. This paper explores the challenges posed by higher-dimensional differential equations in numerical solutions. It highlights the infeasibility of finite difference methods in such scenarios and emphasizes the need for innovative numerical techniques capable of efficiently handling the complexities of higher-dimensional differential equations. Overcoming these challenges is crucial for advancing our understanding and modeling capabilities in complex real-world systems governed by differential equations. Continued research efforts strive to develop novel numerical methodologies capable of addressing these challenges, aiming to broaden the scope of solvable higher-dimensional differential equations and expand their application across diverse scientific domains.

Hybrid methods for solving structural geometric nonlinear dynamic equations: Implementation of fifth-order iterative procedures within composite time integration methods

This paper studies a class of hybrid methods that implement multi-point iterative procedures as the nonlinear solver within optimized composite implicit methods. These multi-point methods are employed to enhance convergence order and accuracy without requiring higher-order derivatives for solving structural geometric nonlinear dynamic equations. The promising approach provides an opportunity to delve deeper into its implications and explore the potential applications of multi-point techniques and composite methods across the analysis of dynamical systems. This study involves the utilization of two-point and five-point iterative methods with fifth-order convergence within three-step and four-step optimized composite implicit time integration methods. Moreover, a comprehensive hybrid scheme is introduced by integrating a family of optimized composite implicit methods with single-point and multi-point iterative methods. Then, the performance of the hybrid scheme is analyzed through structural examples, considering factors such as time step size, tolerance threshold, and the degrees of freedom of structures. The findings and numerical results illustrate that multi-point methods outperform the single-point method in terms of the number of iterations. Simultaneously, the four-step time integration method exhibits superior performance compared to the three-step time integration method, albeit with a marginal difference. It is observed that the hybrid method, consisting of the four-step time integration method and the five-point iterative method, performs better than other methods in terms of the number of iterations and errors for solving geometric nonlinear dynamic equations, especially in large structures. Thus, it stands out as a favorable choice for practical analyses and a possible candidate for generalization to other types of nonlinear problems in computational mechanics.

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

Analysis and estimation of time step sizes and stiffness parameters for efficient simulations of macromolecules with hydrodynamic interactions

Recent studies have shown the importance of using highly resolved models for Brownian Dynamics (BD) simulations of long macromolecules. For computational efficiency, such models use stiff springs to mimic a single Kuhn step and use a single-step semi-implicit (SS) scheme. Somewhat unexpectedly, time step sizes for such a single-step method need to be reduced with increasing chain size and level of hydrodynamic interactions (HIs), for good convergence. The conventional predictor–corrector (PC) method works reasonably well but remains computationally slow, owing to multiple iterations per time step to convergence. In this study, we reveal how the time step size for the much faster SS method is tied to the physics of the problem. Using simple physical principles, we derive an analytical estimate of the upper limit on the time step size for given levels of HI, chain size, and stiffness of connecting springs. Detailed BD simulations at equilibrium and in flow fields highlight the success of our analytical estimate. We also provide a lower limit on spring stiffness parameter such that it remains effectively rigid and successfully mimics a Kuhn step. Our investigations show that the resulting BD simulations using our estimated time step size in the SS scheme are significantly faster than the conventional PC technique. The analysis presented here is expected to be useful in general for any type of simulations of macromolecules, with or without flow fields, owing to deep connections with the underlying physics.

The generalized Kelvin chain-based model for an orthotropic viscoelastic material

We propose a constitutive material model to describe the rheological (viscoelastic) mechanical response of timber. The viscoelastic model is based on the generalized Kelvin chain applied to the orthotropic material and is compared to the simple approach given by standards. The contribution of this study consists of the algorithmization of the viscoelastic material model of the material applied to the orthotropic constitutive law and implementation into the FEM solver. In the next step, the fitting of the input parameters of the Kelvin chain is described, and at least a material model benchmark and comparison to the approach given by standards were done. The standardized approach is based on the reduction of the material rigidity at the end of the loading period using a creep coefficient, whereas the loading history state variables are not considered when establishing the result for a specific time step. The paper presents the benefits of the rheological model. It also demonstrates the fitting algorithm based on particle swarm optimization and the least squares method, which are essential for the use of the generalized Kelvin chain model. The material model based on the orthotropic generalized Kelvin chain was implemented into the FEM solver for the shell elements. This material model was validated on the presented benchmark tasks, and the influence of the time step size on the accuracy of model results was analyzed.

Energy diminishing implicit-explicit Runge–Kutta methods for gradient flows

This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge–Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.

The stabilized exponential-SAV approach for the Allen–Cahn equation with a general mobility

In this paper, we construct a second-order accurate, energy stable and maximum bound principle-preserving scheme for the Allen–Cahn equation with a general mobility based on the stabilized exponential scalar auxiliary variable (SESAV) approach. Some extra stabilizing terms are added to the discretized scheme for the purpose of improving numerical stability. We first proved the maximum bound principle (MBP) under reasonable constraints on time step size and the stabilization parameter. Then, we found that the proposed scheme is unconditionally energy-stable. Finally, a numerical example is carried out to verify the theoretical results.

A novel approach to solve hyperbolic Buckley-Leverett equation by using a transformer based physics informed neural network

Solving hyperbolic partial differential equation is a challenging task due to the non-linear feature that requires to capture the shock wave. Numerical solution relies on discretization of both spatial and temporal domain, and iterative approach like Newton's method is involved and time step size is crucial for stability and convergence in the presence of non-linearity. Physics-informed neural networks (PINNs) offer a new and versatile approach for solving partial different equations by minimizing the residual of governing equations and approaching to the initial and boundary conditions. Currently, most PINNs are built based on a simple fully connected neural network which exhibits some limitations to model complex non-linear partial differential equations. In this paper, a novel method is developed to combine Transformer model and PINNs approach (Tr-PINN) to solving a hyperbolic partial differential equation directly without any prior knowledge. Tr-PINN method is based on a series of Transformer blocks where self-attention mechanism is used to capture the non-linearity features of the solution. Unlike most PINNs models generate inputs with spatial and temporal vectors only, Tr-PINN introduces the non-linearity term mobility ratio as additional input vector. The method is tested on a classical hyperbolic problem, called Buckley-Leverett equation with non-convex flux function. We found that the Tr-PINN method can capture the water shock front effectively and provide a general solution for Buckley-Leverett equation under various mobility ratio conditions.