Adaptive Time-stepping Scheme Research Articles

Jacobian-free Locally Linearized Runge–Kutta method of Dormand and Prince for large systems of differential equations

This paper introduces Jacobian-free Locally Linearized Runge–Kutta formulas of Dormand and Prince for integrating large systems of initial value problems. With these new Jacobian-free formulas, an adaptive time-stepping variable-order scheme with regulated Krylov dimension is constructed with new developments in the computation of the phi-function action over vectors through Jacobian-free Krylov–Padé approximations, and in the procedures for controlling the approximation errors and for estimating the dimension of the Krylov subspaces. At each integration step, the implemented scheme performs only one Krylov subspace decomposition and a few computations of low dimensional exponential matrices, which contrasts with the implementation of other exponential-type integrators of high order. Regarding to existing schemes, the performed numerical study demonstrates a higher effectiveness of the constructed scheme in the integration of a variety of test equations.

Strong convergence of adaptive time-stepping schemes for the stochastic Allen–Cahn equation

Abstract It is known from Beccari et al. (2019) that the standard explicit Euler-type scheme (such as the exponential Euler and the linear-implicit Euler schemes) with a uniform timestep, though computationally efficient, may diverge for the stochastic Allen–Cahn equation. To overcome the divergence, this paper proposes and analyzes adaptive time-stepping schemes, which adapt the timestep at each iteration to control numerical solutions from instability. The a priori estimates in $\mathscr{C}(\mathscr{O})$-norm and $\dot{H}^{\beta }(\mathscr{O})$-norm of numerical solutions are established provided the adaptive timestep function is suitably bounded, which plays a key role in the convergence analysis. We show that the adaptive time-stepping schemes converge strongly with order $\frac{\beta }{2}$ in time and $\frac{\beta }{d}$ in space with $d$ ($d=1,2,3$) being the dimension and $\beta \in (0,2]$. Numerical experiments show that the adaptive time-stepping schemes are simple to implement and at a lower computational cost than a scheme with the uniform timestep.

Convergence analysis of the maximum principle preserving BDF2 scheme with variable time-steps for the space fractional Allen–Cahn equation

In this work, a fully implicit discrete scheme preserving the maximum principle is analyzed for solving the space fractional Allen–Cahn equation with variable step BDF2 in time. The convergence analysis in L∞ norm is rigorously proved under the same requirement of time-step ratios for the maximum principle by means of the kernels recombination technique. Furthermore, the unique solvability of the proposed scheme is theoretically guaranteed depending on the property of convex functions. The energy decay law with the consideration of Riesz fractional derivative is also proved under sufficient restrictions on time-steps and time-step ratios. Numerical experiments are given to validate the second order accuracy in time, the maximum principle as well as the energy decay law of the proposed numerical scheme. In particular, comparisons of errors and time levels for the adaptive time-stepping strategy and uniform time steps are conducted to demonstrate the efficiency of adaptive time-stepping strategy.

High-order energy stable variable-step schemes for the time-fractional Cahn–Hilliard model

Based on the Alikhanov formula of the Caputo derivative and the exponential scalar auxiliary variable approach, two different second-order stable numerical schemes are constructed for the time-fractional Cahn–Hilliard model, including the nonlinear L2-1σ scheme and the linear L2-1σ-ESAV scheme. Applying the discrete gradient structure, we construct the asymptotically compatible energies and the associated energy dissipation laws of these two schemes for the time-fractional Cahn–Hilliard model. Several experiments are carried out to demonstrate the accuracy of our schemes and compare their computational efficiency, as well as to investigate the coarsening dynamics of the time-fractional Cahn–Hilliard model with different adaptive time-stepping strategies. In the long-time simulations, the linear L2-1σ-ESAV scheme may be more costly than the nonlinear L2-1σ scheme although the linear scheme may take less time at each time level. It is observed that the energy of the L2-1σ-ESAV scheme prones to produce small fluctuations when large time steps are used.

Adaptive time-step unscented kalman filtering (ATS-UKF) based observer design for urea selective catalytic reduction (SCR) performance of diesel engines

To reduce the number of sensors in the SCR catalyst, state feedback and fault diagnosis information are provided. Firstly, a model based on the coupling of flow, heat transfer, and gas-solid phase catalytic reaction in the SCR system is investigated in this paper. The parabolic partial differential equations are simplified by the variable substitution method and the method of lines approach (MOL). The simplified system of equations is solved by backward differentiation formulas (BDF) with adaptive adjustment time step strategy. Meanwhile, the chemical reaction parameters are accurately calibrated per second using the Levenberg-Marquardt method. Secondly, the ATS-UKF is designed in this paper, and to ensure the synchronisation between the ATS-UKF and the SCR model calculations, the time step of solving the BDF by the SCR model is taken as the time step of propagating the sigma points. Two observation scenarios are assumed: (1) no downstream NH3 concentration sensor, ammonia coverage and downstream NH3 concentration are observed by ATS-UKF; (2) no downstream NOx sensor, ammonia coverage and downstream NOx concentration are observed by ATS-UKF. Finally, the paper carries out bench tests. In the first case, the ammonia coverage obtained by the ATS-UKF reached 0.99 with respect to the model-calculated value R². The mean absolute error (MAE) between the observed and experimental values of the ATS-UKF for the downstream NH3 concentration was 2.76 ppm. In the second case, the ammonia coverage obtained by the ATS-UKF reached 0.99 with respect to the model-calculated value R², and the MAE between the observed and experimental values of the ATS-UKF for the downstream NOx concentration was 1.53 ppm. Environmental ImplicationThe Adaptive Time-Step Unscented Kalman Filtering (ATS-UKF) enhances urea Selective Catalytic Reduction (SCR) in diesel engines, improving environmental outcomes. This method minimizes sensor dependence, enabling more precise SCR system management and effective emission reduction. By advancing emission control technologies, ATS-UKF contributes to global air pollution mitigation efforts, supporting cleaner air and environmental sustainability. Its innovative approach in monitoring and predicting SCR performance marks a significant step towards eco-friendly diesel engine operation.

Second-order linear adaptive time-stepping schemes for the fractional Allen–Cahn equation

Second-order linear adaptive time-stepping schemes for the fractional Allen–Cahn equation

A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier–Stokes equations

We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxiliary variable (SAV) scheme for Navier–Stokes equation with periodic boundary conditions (Huang and Shen, SIAM J. Numer. Anal. 59 (2021) 2926–2954) to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key ingredients of our analysis are a new modified discrete Grönwall inequality, exploration of the discrete orthogonal convolution (DOC) kernels, and the unconditional stability of the proposed scheme. We derive global and local optimal H1 error estimates in 2D and 3D, respectively. Our analysis provides a theoretical support for solving Navier–Stokes equations using variable time-step IMEX-BDF2 SAV schemes. We also design an adaptive time-stepping strategy, and provide ample numerical examples to confirm the effectiveness and efficiency of our proposed methods.

Sharp L2 Norm Convergence of Variable-Step BDF2 Implicit Scheme for the Extended Fisher–Kolmogorov Equation

A variable-step BDF2 time-stepping method is investigated for simulating the extended Fisher-Kolmogorov equation. The time-stepping scheme is shown to preserve a discrete energy dissipation law if the adjacent time-step ratios r n ≔ Τ n / Τ n − 1 < 3 + 17 / 2 ≈ 3.561 . With the aid of discrete orthogonal convolution kernels, concise L 2 norm error estimates are proved, for the first time, under the mild step ratios constraint 0 < r n < 3.561 . Our error estimates are almost independent of the step ratios r n so that the proposed numerical scheme is robust with respect to the variations of time steps. An adaptive time-stepping strategy based on solution accuracy is then applied to update the computational efficiency. Numerical examples are included to illustrate our theoretical results.

FLEKS: A flexible particle-in-cell code for multi-scale plasma simulations

The magnetohydrodynamics with embedded particle-in-cell (MHD-EPIC) model has been successfully applied to global magnetospheric simulations in recent years. However, the PIC region was restricted to be one or more static boxes, which is not always sufficient to cover the whole physical structure of interest efficiently. The FLexible Exascale Kinetic Simulator (FLEKS), which is a new PIC code and allows a dynamic PIC region of any shape, is designed to break this restriction. FLEKS is usually used as the PIC component of the MHD with adaptively embedded particle-in-cell (MHD-AEPIC) model. FLEKS supports dynamically activating or deactivating cells to fit the regions of interest during a simulation. An adaptive time-stepping scheme is also introduced to improve the accuracy and efficiency of a long simulation. The particle number per cell may increase or decrease significantly and lead to load imbalance and large statistical noise in the cells with fewer particles. A particle splitting scheme and a particle merging algorithm are designed to limit the change of the particle number and hence improve the accuracy of the simulation as well as load balancing. Both particle splitting and particle merging conserve the total mass, momentum, and energy. FLEKS also contains a test-particle module to enable tracking particle trajectories due to the time-dependent electromagnetic field that is obtained from a global simulation.

Numerical Study on the Water Entry of a Freely Falling Unmanned Aerial-Underwater Vehicle

The unmanned aerial–underwater vehicle (UAUV) is a new type of vehicle that can fly in the air and cruise in water and is expected to cross the free water surface several times to perform continuous uninterrupted observation and sampling. To analyze the hydrodynamic and motion characteristics of the vehicle, the whole water-entry process of a multi-degree-of-freedom UAUV with various velocity and pitch angle was investigated through a Reynolds-averaged Navier–Stokes method. The computational domain was meshed by trimmer cells. The relative movement between the vehicle and fluid domain was simulated using moving reference frame overset mesh to delineate the interaction region around vehicle body. To reduce the computational cost, adaptive mesh refinement and adaptive time-stepping strategy were used to capture the slamming pressure accurately with reasonable computational effort. First, convergence study is considered. Simulations of the vehicle with various initial velocities and different pitch angles were performed. The variable physical properties were analyzed, and detailed results through the time-varying force and velocity were shown. Initial velocity and pitch angle are found to significantly influence hydrodynamic behavior, including the time-varying force, while thickness ratio has a great impact on added mass and pressure. The results show that higher entry velocity results in greater peak vertical force. The transverse hydrodynamic characteristics for oblique water entry of the vehicle with varies pith angle are quite different.

Transient characteristics of a solid oxide electrolysis cell under different voltage ramps: Transport phenomena behind overshoots

A promising solution to the storage of intermittent renewable energy is to integrate solid oxide electrolysis cells (SOEC) with solar/wind power. This trend necessitates comprehensive, quantitative investigations on the transient characteristics of SOEC, especially under varying power-supply conditions. For this purpose, a high-resolution, 3-dimensional, transient numerical model, as well as an adaptive time-stepping strategy, is proposed in this study. This study analyzes the electrical, gaseous, and thermal responses of SOEC to voltage ramps with different ramp rates and ramp magnitudes. The results show that electrical undershoots or overshoots occur after fast voltage changes. This phenomenon reflects the discrepancies between the steady and transient current–voltage characteristics and may lead to unsteady hydrogen production rates in practice. The electrical undershoots or overshoots are caused by the different transfer rates in SOEC – electronic/ionic transfer rate is faster than mass transfer rate, and mass transfer rate is faster than heat transfer rate. Furthermore, the electrical undershoots or overshoots can be divided into two parts. One part is related to mass-transfer lag, and the other part is related to heat-transfer lag. The former can be alleviated or eliminated by simply slowing down the voltage ramp, while the latter needs a more effective control strategy other than merely adjusting the voltage ramps. Apart from the electrical conditions, cell structure also has significant impacts on the electrical responses, e.g., the rib and the length of channel are related to the non-uniform electrical responses in the functional layer. Finally, via a quantitative technique developed from linear time-invariant systems, it is shown that the electrical responses of SOEC are governed by two time constants in the functional layer, namely the mass-transfer time constant (estimated as τm,H2O,FL=0.00723s) and the heat-transfer time constant (estimated as τt,FL=180s).

Energy stability of a temporal variable-step difference scheme for time-fractional nonlinear fourth-order reaction-diffusion equation

In this paper, we propose a nonuniform numerical formula for the Caputo fractional derivative at the half-grid based on the piecewise linear interpolation and construct a difference scheme for the time-fractional nonlinear fourth-order reaction-diffusion equation. By virtue of two discrete tools: the discrete orthogonal convolution kernels and the discrete complementary convolution kernels, we obtain the positive definiteness of the discrete time-fractional derivative. Then a discrete variational energy dissipation law of the proposed difference scheme is established for the time-fractional nonlinear fourth-order reaction-diffusion equation, which is asymptotically compatible with the associated energy dissipation law for the classical equation as the value of fractional order approaches to one. Numerical experiments demonstrate the effectiveness and the energy dissipation of the proposed difference scheme with an adaptive time-stepping strategy.

An Implicit–Explicit Second-Order BDF Numerical Scheme with Variable Steps for Gradient Flows

In this paper, we propose and analyze an efficient implicit–explicit second-order backward differentiation formulation (BDF2) scheme with variable time steps for gradient flow problems using a scalar auxiliary variable (SAV) approach. Comparing with the traditional second-order SAV approach (Shen et al. in J Comput Phys 353:407–416, 2018), we only use a first-order method to approximate the auxiliary variable. This treatment does not affect the second-order accuracy of the unknown function $$\phi $$ , and is essentially important for deriving the unconditional energy stability of the proposed BDF2 scheme with variable time steps. We prove the unconditional energy stability of the scheme for a modified discrete energy with the adjacent time step ratio $$\gamma _{n+1}:=\tau _{n+1}/\tau _{n}\le 4.8645$$ . The uniform $$H^{2}$$ bound for the numerical solution is derived under a mild regularity restriction on the initial condition, that is $$\phi ({\varvec{x}},0)\in H^{2}$$ . Based on this uniform bound of the numerical solution, a rigorous error estimate of the proposed scheme is carried out on the nonuniform temporal mesh. Finally, serval numerical tests are provided to validate the theoretical claims. With the application of an adaptive time-stepping strategy, the efficiency of our proposed scheme can be clearly observed in the coarsening dynamics simulation.

An integral equation method for the advection-diffusion equation on time-dependent domains in the plane

Boundary integral methods are attractive for solving homogeneous elliptic partial differential equations on complex geometries, since they can offer accurate solutions with a computational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to apply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adaptive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution methodology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy.

A maximum bound principle preserving iteration technique for a class of semilinear parabolic equations

This paper presents a systematic methodology, called the MBP-preserving iteration technique, to develop the maximum bound principle (MBP) preserving numerical algorithms for a class of semilinear parabolic equations. Two types of MBP-preserving iterations are suggested to solve two well-known θ-weighted schemes, respectively. Within some mild time-step constraints, the corresponding iteration solutions are proved to preserve the MBP property at each iteration step so that the numerical scheme has a uniquely MBP-preserving solution. In addition, concise error estimates in the maximum norm are established on nonuniform time meshes. Several numerical examples coupled with an adaptive time-stepping strategy are implemented for the Allen-Cahn model to confirm the theoretical findings and demonstrate their effectiveness for long-time numerical simulations.

An efficient bound-preserving and energy stable algorithm for compressible gas flow in porous media

Due to the great significance of the natural gas and shale gas, it is becoming increasingly important to simulate compressible gas flow in porous media. The model obeys an energy dissipation law as well as molar density must be positive and bounded in terms of the equation of state. For the purpose of eliminating nonphysical solutions as well as improving the stability in practical simulation, preservation of these properties is essential for a promising numerical method, but it is actually challenging due to the strong nonlinearity and complexity of the model. In this paper, we propose an efficient linearized numerical scheme that inherits the energy dissipation law as well as preserves the boundedness of molar density. Specifically, to treat the logarithmic type Helmholtz free energy density determined by the Peng-Robinson equation of state, we propose a novel adaptive stabilization approach involving the second derivatives of the convex energy terms. At each time step, the stabilization parameter is adaptively updated by a simple and explicit formula to ensure the energy dissipation law. The stabilized and linearized chemical potential allows to formulate the local mass conservation equation as an equivalent convection-diffusion form, and from this, an adaptive time step strategy is proposed to preserve the positivity and boundedness of molar density. The calculation of the time step size is fully explicit and easy to implement. Additionally, the fully discrete scheme is constructed using the conservative cell-centered finite difference method with the upwind strategy, and thus, it enjoys the local mass conservation. Numerical results are also presented to demonstrate the excellent performance of the proposed scheme.

A structure-preserving and variable-step BDF2 Fourier pseudo-spectral method for the two-mode phase field crystal model

For the two-mode phase field crystal models, the evolutions of the solutions and energy vary fast at certain time. To resolve varying time scales efficiently and reduce the computational cost, a variable-step BDF2 Fourier pseudo-spectral method is proposed. It is shown that the fully-discrete scheme is volume-conserving and unconditional energy-stable. Moreover, a robust error estimate is established by using the discrete orthogonal convolution kernels and the corresponding convolution inequalities. Numerical experiments by using the random and adaptive time-stepping strategies are presented to confirm the effectiveness of the scheme.

A second-order energy stable and nonuniform time-stepping scheme for time fractional Burgers' equation

In this paper, an effective finite difference scheme of high order accuracy is proposed for the nonlinear time fractional Burgers' equation. Specifically, we apply the Alikhanov's scheme on graded mesh in the temporal direction and a novel fourth-order compact scheme in the spatial discretization. The proposed scheme resolves initial weak singularity of the solution and preserves high resolution in the space direction. It is rigorously proved that the finite difference scheme is uniquely solvable, discrete variational energy dissipation law and unconditionally stable and convergent in sense of discrete L 2 -norm. With appropriate choice of the grading parameter, the convergence accuracy is min ⁡ { r α , 2 } order in time and fourth order in space, where r is the mesh grading. In the numerical implementation procedure, fast convolution technique and adaptive time-stepping strategy are adopted to accelerate the presented solver and to capture evolution of the solution. Numerical experiments are carried out to verify the validity and effectiveness of the proposed scheme for solving nonlinear time-fractional Burgers' equation.

A new implementation of the geometric method for solving the Eady slice equations

We present a new implementation of the geometric method of Cullen & Purser (1984) for solving the semi-geostrophic Eady slice equations, which model large scale atmospheric flows and frontogenesis. The geometric method is a Lagrangian discretisation, where the PDE is approximated by a particle system. An important property of the discretisation is that it is energy conserving. We restate the geometric method in the language of semi-discrete optimal transport theory and exploit this to develop a fast implementation that combines the latest results from numerical optimal transport theory with a novel adaptive time-stepping scheme. Our results enable a controlled comparison between the Eady-Boussinesq vertical slice equations and their semi-geostrophic approximation. We provide further evidence that weak solutions of the Eady-Boussinesq vertical slice equations converge to weak solutions of the semi-geostrophic Eady slice equations as the Rossby number tends to zero.

Asymptotically compatible energy law of the Crank–Nicolson type schemes for time-fractional MBE models

This paper is concerned with the numerical simulations for the dynamics of the time-fractional molecular beam epitaxy models. The variable-step Crank–Nicolson-type schemes are proposed and analyzed for model with and without slope selection respectively. By using the discrete gradient structures of discrete fractional derivative, the numerical schemes preserve discrete variational energy dissipation law on general meshes unconditionally. These are generalizations of classical energy dissipation laws and are asymptotically compatible with the classical MBE model. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our schemes.