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Exponential/extended Runge–Kutta–Nyström (ERKN) methods have been validated by numerous theoretical and numerical results to be more efficient and suitable than classical Runge–Kutta–Nyström (RKN) methods in dealing with highly oscillatory Hamiltonian systems. Once numerical solutions are required to achieve a prescribed local error tolerance for practical problems, the embedded ERKN pairs with adaptive stepsize are needed. Given the higher efficiency of high-order methods than low-order methods, this paper focuses on high-order embedded ERKN pairs. Using the particular mapping from RKN methods into ERKN methods, we establish an approach to constructing high-order embedded ERKN pairs. By diagonalizing the frequency matrix, an implementation algorithm with less calculation amount than the direct calculation procedure is proposed for high-dimensional problems. In addition, the dispersion and dissipation of the ERKN pairs ERKN4(3) and ERKN8(6) are analyzed. Furthermore, the application of ERKN pairs to an important highly oscillatory problem, i.e., the wave equation prescribed with different boundary conditions is discussed. For the periodic boundary condition, a special implementation algorithm incorporated with FFT techniques is presented, which decreases the calculation amount in one time step from O (N 2) to O (N log ⁡ N). Finally, numerical results with the FPU problem, the Klein–Gordon equation in the nonrelativistic limit regime, and a two-dimensional wave equation demonstrate the superiority of ERKN pairs over classical RKN pairs. • An effective approach is presented to establish high-order embedded ERKN pairs. • A special implementation algorithm by diagonalizing the frequency matrix is proposed to reduce the calculation amount. • The application of the embedded ERKN pairs to wave equations prescribed with different boundary conditions is discussed. [ABSTRACT FROM AUTHOR]

We propose an easy-to-implement iterative method for resolving the implicit (or semi-implicit) schemes arising in solving reaction-diffusion (RD) type equations. We formulate the nonlinear time implicit scheme as a min-max saddle point problem and then apply the primal-dual hybrid gradient (PDHG) method. Suitable precondition matrices are applied to the PDHG method to accelerate the convergence of algorithms under different circumstances. Furthermore, our method is applicable to various discrete numerical schemes with high flexibility. From various numerical examples tested in this paper, the proposed method converges properly and can efficiently produce numerical solutions with sufficient accuracy. [ABSTRACT FROM AUTHOR]

• A linearly operator splitting algorithm is proposed for the nonlocal VCH equation. • The energy stabilities for both subproblems are proved. • The stability and convergence of the operator splitting algorithm are studied. • A semi-implicit SDC method is further used to improve time accuracy. Recently, the viscous Cahn-Hilliard (VCH) equation has been proposed as a phenomenological continuum model for phase separation in glass and polymer systems where intermolecular friction forces become important. Compared with the classical local VCH model, the nonlocal VCH model equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions of microstructures in materials. This paper presents a high order fast explicit method based on operator splitting and spectral deferred correction (SDC) for solving the nonlocal VCH equation. We start with a second-order operator splitting spectral scheme, which is based on the Fourier spectral method and the strong stability preserving Runge-Kutta (SSP-RK) method. The scheme takes advantage of applying the fast Fourier transform (FFT) and avoiding nonlinear iteration. The stability and convergence analysis of the obtained numerical scheme are analyzed. To improve the temporal accuracy, the semi-implicit SDC method is then introduced. Various numerical simulations are performed to validate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

• A new algorithm incorporates model reduction in the spatial and temporal domains. • We study parabolic problems with heterogeneous coefficients and rough initial data. • We derive convergence analysis that weakly depends on the heterogeneous coefficients. • The convergence is rigorously studied, which greatly improves the current result. • Extensive numerical tests are performed to show the fast convergence of our algorithm. We propose in this paper the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm to solve parabolic equations with heterogeneous coefficients efficiently. This algorithm combines the advantages of multiscale methods that can deal with heterogeneity in the spatial domain effectively, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. We derive the convergence rate of this algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step and the fine-scale time step. Extensive numerical tests are presented to demonstrate the performance of our algorithm, which verify our theoretical results perfectly. [ABSTRACT FROM AUTHOR]

• A neural network enhanced Boltzmann model is proposed. • The mechanical and neural models are unified into a differentiable architecture and the neural- ODE-type training strategy is constructed. • A general numerical scheme is designed to solve the universal Boltzmann equation. • Numerical experiments of homogeneous and inhomogeneous systems are provided to validate the current method. One of the biggest challenges for simulating the Boltzmann equation is the evaluation of fivefold collision integral. Given the recent successes of deep learning and the availability of efficient tools, it is an obvious idea to try to substitute the calculation of the collision operator by the evaluation of a neural network. However, it is unlcear whether this preserves key properties of the Boltzmann equation, such as conservation, invariances, the H-theorem, and fluid-dynamic limits. In this paper, we present an approach that guarantees the conservation properties and the correct fluid dynamic limit at leading order. The concept originates from a recently developed scientific machine learning strategy which has been named "universal differential equations". It proposes a hybridization that fuses the deep physical insights from classical Boltzmann modeling and the desirable computational efficiency from neural network surrogates. The construction of the method and the training strategy are demonstrated in detail. We conduct an asymptotic analysis and illustrate the multi-scale applicability of the method. The numerical algorithm for solving the neural network-enhanced Boltzmann equation is presented as well. Several numerical test cases are investigated. The results of numerical experiments show that the time-series modeling strategy enjoys the training efficiency on this supervised learning task. [ABSTRACT FROM AUTHOR]

• First dynamical low-rank algorithm that is mass, momentum, and energy conservative. • Can be combined with an explicit integrator that maintains conservation. • Conserves the underlying continuity equations in addition to the invariants. • Low-rank breaks the curse of dimensionality for high-dimensional kinetic equations. The primary challenge in solving kinetic equations, such as the Vlasov equation, is the high-dimensional phase space. In this context, dynamical low-rank approximations have emerged as a promising way to reduce the high computational cost imposed by such problems. However, a major disadvantage of this approach is that the physical structure of the underlying problem is not preserved. In this paper, we propose a dynamical low-rank algorithm that conserves mass, momentum, and energy as well as the corresponding continuity equations. We also show how this approach can be combined with a conservative time and space discretization. [ABSTRACT FROM AUTHOR]