We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes of fully-discrete conservative methods for several nonlinear dispersive wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations conserve all linear invariants and one nonlinear invariant for each system. The spatial semidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially explicit, using relaxation Runge-Kutta methods. We implement some specific schemes from among the derived classes, and demonstrate their favorable properties through numerical tests.

In this paper, the second-order and third-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (WENO) limiters are proposed on tetrahedral meshes. The multi-resolution WENO limiter is an extension of a finite volume multi-resolution WENO scheme developed in [81], which serves as a limiter for RKDG methods on tetrahedral meshes. This new WENO limiter uses information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF indicator [42], to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the second or third degree of the DG solution. The second-order and third-order RKDG methods with the associated multi-resolution WENO limiters are developed as examples for general high-order RKDG methods, which could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property near strong discontinuities by gradually degrading from the optimal order to the first order. The linear weights inside the procedure of the new multi-resolution WENO limiters can be set as any positive numbers on the condition that they sum to one. A series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell on tetrahedral meshes. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy on tetrahedral meshes. Such spatial reconstruction methodology improves the robustness in the simulation on the same compact spatial stencil of the original DG methods on tetrahedral meshes. Extensive one-dimensional (run as three-dimensional problems on tetrahedral meshes) and three-dimensional tests are performed to demonstrate the good performance of the RKDG methods with new multi-resolution WENO limiters.

The Riemann solver with internal reconstruction (RSIR) of Carmouze et al. (2020) is investigated, revisited and improved for the Euler equations. In this reference, the RSIR approach has been developed to address the numerical resolution of the non-equilibrium two-phase flow model of Saurel et al. (2017). The main idea is to reconstruct two intermediate states from the knowledge of a simple and robust intercell state such as HLL, regardless the number of waves present in the Riemann problem. Such reconstruction improves significantly the accuracy of the HLL solution, preserves robustness and maintains stationary discontinuities. Consequently, when dealing with complex flow models, such as the aforementioned one, RSIR-type solvers are quite easy to derive compared to HLLC-type ones that may require a tedious analysis of the governing equations across the different waves. In the present contribution, the RSIR solver of Carmouze et al. (2020) is investigated, revisited and improved with the help of thermodynamic considerations, making a simple, accurate, robust and positive Riemann solver. It is also demonstrated that the RSIR solver is strictly equivalent to the HLLC solver of Toro et al. (1994) for the Euler equations when the Rankine-Hugoniot relations are used. In that sense, the RSIR approach recovers the HLLC solver in some limit as well as the HLL one in another limit and can be simplified at different levels when complex systems of equations are addressed. For the sake of clarity and simplicity, the derivations are performed in the context of the one-dimensional Euler equations. Comparisons and validations against the conventional HLLC solver and exact solutions are presented.

We propose a coupled model to simulate shallow water waves induced by elastic deformations in the bed topography. The governing equations consist of the depth-averaged shallow water equations including friction terms for the water free-surface and the well-known second-order elastostatics formulation for the bed deformation. The perturbation on the free-surface is assumed to be caused by a sudden change in the bottom beds. At the interface between the water flow and the bed topography, transfer conditions are implemented. Here, the hydrostatic pressure and friction forces are considered for the elastostatic equations whereas bathymetric forces are accounted for in the shallow water equations. The focus in the present study is on the development of a simple and accurate representation of the interaction between water waves and bed deformations in order to simulate practical shallow water flows without relying on complex partial differential equations with free boundary conditions. The effects of location and magnitude of the deformation on the flow fields and free-surface waves are investigated in details. Numerical simulations are carried out for several test examples on shallow water waves induced by sudden changes in the bed. The proposed computational model has been found to be feasible and satisfactory.

The quad-curl problem arises in the resistive magnetohydrodynamics (MHD) and the electromagnetic interior transmission problem. In this paper we study a new mixed finite element scheme using Nédélec's edge elements to approximate both the solution and its curl for quad-curl problem on Lipschitz polyhedral domains. We impose element-wise stabilization instead of stabilization along mesh interfaces. Thus our scheme can be implemented as easy as standard Nédélec's methods for Maxwell's equations. Via a discrete energy norm stability due to element-wise stabilization, we prove optimal convergence under a low regularity condition. We also extend the mixed finite element scheme to the quad-curl eigenvalue problem and provide corresponding convergence analysis based on that of source problem. Numerical examples are provided to show the viability and accuracy of the proposed method for quad-curl source problem.

Gaussian processes and other kernel-based methods are used extensively to construct approximations of multivariate data sets. The accuracy of these approximations is dependent on the data used. This paper presents a computationally efficient algorithm to greedily select training samples that minimize the weighted $L^p$ error of kernel-based approximations for a given number of data. The method successively generates nested samples, with the goal of minimizing the error in high probability regions of densities specified by users. The algorithm presented is extremely simple and can be implemented using existing pivoted Cholesky factorization methods. Training samples are generated in batches which allows training data to be evaluated (labeled) in parallel. For smooth kernels, the algorithm performs comparably with the greedy integrated variance design but has significantly lower complexity. Numerical experiments demonstrate the efficacy of the approach for bounded, unbounded, multi-modal and non-tensor product densities. We also show how to use the proposed algorithm to efficiently generate surrogates for inferring unknown model parameters from data using Bayesian inference.

In this paper, we propose, analyze, and numerically validate an adaptive finite element method for the Cahn–Hilliard–Navier–Stokes equations. The adaptive method is based on a linear, decoupled scheme introduced by Shen and Yang [30]. An unconditionally energy stable discrete law for the modified energy is shown for the fully discrete scheme. A superconvergent cluster recovery based a posteriori error estimations are constructed for both the phase field variable and velocity field function, respectively. Based on the proposed space and time discretization error estimators, a time-space adaptive algorithm is designed for numerical approximation of the Cahn–Hilliard–Navier–Stokes equations. Numerical experiments are presented to illustrate the reliability and efficiency of the proposed error estimators and the corresponding adaptive algorithm.

The presented work is dedicated to the mathematical and numerical modeling of unsteady single- and two-phase flows using finite volume and penalty methods. Two classes of Navier-Stokes solvers are considered. Their accuracy and robustness are compared to identify their respective strengths and weaknesses. Exact (also referred to as monolythic) solvers such as the Augmented Lagrangian and the Fully Coupled methods address the saddle-point structure on the pressure-velocity couple of the discretized system by means of a penalization term or even directly, whereas approximate (segregated) solvers such as the Standard Projection method rely on operator splitting to break the problem down to decoupled systems. The objective is to compare all approaches in the context of two-phase flows at high viscosity and density ratios. To characterize the interface location, a volume-of-fluid (VOF) approach is used based on a Piecewise Linear Interface Construction (PLIC). Various 2D simulations are performed on single- and two-phase flows to characterize the behavior and performances of the various solvers.

Atomistic/continuum coupling method is a class of multiscale computational method for the efficient simulation of crystalline defects. The recently developed blended ghost force correction (BGFC) method combines the efficiency of blending methods and the accuracy of quasi-nonlocal (QNL) type methods. BGFC method can be applied to multi-body interaction potentials and general interfaces. In this paper, we present the formulation, implementation and analysis of the BGFC method in three dimensions. In particular, we focus on the difference and connection with other blending variants, such as energy based blended quasi-continuum method (BQCE) and force based blended quasi-continuum method (BQCF). The theoretical results are justified by a few benchmark numerical experiments with point defects and microcrack in the three dimensional face-centered cubic (FCC) lattice.

Data assimilation is a technique for increasing the accuracy of simulations of solutions to partial differential equations by incorporating observable data into the solution as time evolves. Recently, a promising new algorithm for data assimilation based on feedback-control at the PDE level has been proposed in the pioneering work of Azouani, Olson, and Titi (2014). The standard version of this algorithm is based on measurement from data points that are fixed in space. In this work, we consider the scenario in which the data collection points move in space over time. We demonstrate computationally that, at least in the setting of the 1D Allen-Cahn reaction diffusion equation, the algorithm converges with significantly fewer measurement points, up to an order or magnitude in some cases. We also provide an application of the algorithm to the estimation of a physical length scale in the case of a uniform static grid.