We propose in this paper a data driven state estimation scheme for generating nonlinear reduced models for parametric families of PDEs, directly providing data-to-state maps, represented in terms of Deep Neural Networks. A major constituent is a sensor-induced decomposition of a model-compliant Hilbert space warranting approximation in problem relevant metrics. It plays a similar role as in a Parametric Background Data Weak framework for state estimators based on Reduced Basis concepts. Extensive numerical tests shed light on several optimization strategies that are to improve robustness and performance of such estimators.

We consider in this paper random batch interacting particle methods for solving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann (PB) equation as the equilibrium, in the external unbounded domain. To justify the simulation in a truncated domain, an error estimate of the truncation is proved in the symmetric cases for the PB equation. Then, the random batch interacting particle methods are introduced which are $\mathcal{O}(N)$ per time step. The particle methods can not only be considered as a numerical method for solving the PNP and PB equations, but also can be used as a direct simulation approach for the dynamics of the charged particles in solution. The particle methods are preferable due to their simplicity and adaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effects and interactions in the particle methods and to describe phenomena beyond the scope of the mean-field equations.

This paper presents a modeling framework—mathematical model and computational framework—to study the response of a plastic material due to the presence and transport of a chemical species in the host material. Such a modeling framework is important to a wide variety of problems ranging from Li-ion batteries, moisture diffusion in cementitious materials, hydrogen diffusion in metals, to consolidation of soils under severe loading-unloading regimes. The mathematical model incorporates experimental observations reported in the literature on how (elastic and plastic) material properties change because of the presence and transport of a chemical species. Also, the model accounts for one-way (transport affects the deformation but not vice versa) and two-way couplings between deformation and transport subproblems. The resulting coupled equations are not amenable to analytical solutions; so, we present a robust computational framework for obtaining numerical solutions. Given that popular numerical formulations do not produce nonnegative solutions, the computational framework uses an optimized-based nonnegative formulation that respects physical constraints (e.g., nonnegative concentrations). For completeness, we also show the effect and propagation of the negative concentrations, often produced by contemporary transport solvers, into the overall predictions of deformation and concentration fields. Notably, anisotropy of the diffusion process exacerbates these unphysical violations. Using representative numerical examples, we discuss how the concentration field affects plastic deformations of a degrading solid. Based on these numerical examples, we also discuss how plastic zones spread because of material degradation. To illustrate how the proposed computational framework performs, we report various performance metrics such as optimization iterations and time-to-solution.

We propose a high order finite difference linear scheme combined with a high order bound preserving maximum-principle-preserving (MPP) flux limiter to solve the incompressible flow system. For such problem with highly oscillatory structure but not strong shocks, our approach seems to be less dissipative and much less costly than a WENO type scheme, and has high resolution due to a Hermite reconstruction. Spurious numerical oscillations can be controlled by the weak MPP flux limiter. Numerical tests are performed for the Vlasov-Poisson system, the 2D guiding-center model and the incompressible Euler system. The comparison between the linear and WENO type schemes, with and without the MPP flux limiter, will demonstrate the good performance of our proposed approach.

The reliability of BiCGStab and IDR solvers for the exponential scheme discretization of the advection-diffusion-reaction equation is investigated. The resulting discretization matrices have real eigenvalues. We consider BiCGStab, IDR$(S),$ BiCGStab$(L)$ and various modifications of BiCGStab, where $S$ denotes the dimension of the shadow space and $L$ the degree of the polynomial used in the polynomial part. Several implementations of BiCGStab exist which are equivalent in exact arithmetic, however, not in finite precision arithmetic. The modifications of BiCGStab we consider are; choosing a random shadow vector, a reliable updating scheme, and storing the best intermediate solution. It is shown that the Local Minimal Residual algorithm, a method similar to the "minimize residual" step of BiCGStab, can be interpreted in terms of a time-dependent advection-diffusion-reaction equation with homogeneous Dirichlet boundary conditions for the residual, which plays a key role in the convergence analysis. Due to the real eigenvalues, the benefit of BiCGStab$(L)$ compared to BiCGStab is shown to be modest in numerical experiments. Non-sparse (e.g. uniform random) shadow residual turns out to be essential for the reliability of BiCGStab. The reliable updating scheme ensures the required tolerance is truly achieved. Keeping the best intermediate solution has no significant effect. Recommendation is to modify BiCGStab with a random shadow residual and the reliable updating scheme, especially in the regime of large Péclet and small Damköhler numbers. An alternative option is IDR($S$), which outperforms BiCGStab for problems with strong advection in terms of the number of matrix-vector products. The MATLAB code used in the numerical experiments is available on GitLab: https://gitlab.com/ChrisSchoutrop/krylov-adr, a C++ implementation of IDR$(S)$ is available in the Eigen linear algebra library: http: //eigen.tuxfamily.org.

The five-equation model of multi-component flows has been attracting much attention among researchers during the past twenty years for its potential in the study of the multi-component flows. In this paper, we employ a second order finite volume method with minmod limiter in spatial discretization, which preserves local extrema of certain physical quantities and is thus capable of simulating challenging test problems without introducing non-physical oscillations. Moreover, to improve the numerical resolution of the solutions, the adaptive moving mesh strategy proposed in [Huazhong Tang, Tao Tang, Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SINUM, 41: 487-515, 2003] is applied. Furthermore, the proposed method can be proved to be capable of preserving the velocity and pressure when they are initially constant, which is essential in material interface capturing. Finally, several classical numerical examples demonstrate the effectiveness and robustness of the proposed method.

Although interest in numerical approximations of the water wave equation grows in recent years, the lack of rigorous analysis of its time discretization inhibits the design of more efficient algorithms. In practice of water wave simulations, the trade-off between efficiency and stability has been a challenging problem. Thus to shed light on the stability condition for simulations of water waves, we focus on a model simplified from the water wave equation of infinite depth. This model preserves two main properties of the water wave equation: non-locality and hyperbolicity. For the constant coefficient case, we conduct systematic stability studies of the fully discrete approximation of such systems with the Fourier spectral approximation in space and general Runge-Kutta methods in time. As a result, an optimal time discretization strategy is provided in the form of a modified CFL condition, i.e. $∆t = \mathcal{O}(\sqrt{∆x}).$ Meanwhile, the energy stable property is established for certain explicit Runge-Kutta methods. This CFL condition solves the problem of efficiency and stability: it allows numerical schemes to stay stable while resolves oscillations at the lowest requirement, which only produces acceptable computational load. In the variable coefficient case, the convergence of the semi-discrete approximation of it is presented, which naturally connects to the water wave equation. Analogue of these results for the water wave equation of finite depth is also discussed. To validate these theoretic observation, extensive numerical tests have been performed to verify the stability conditions. Simulations of the simplified hyperbolic model in the high frequency regime and the water wave equation are also provided.

We propose a new high order accurate nodal discontinuous Galerkin (DG) method for the solution of nonlinear hyperbolic systems of partial differential equations (PDE) on unstructured polygonal Voronoi meshes. Rather than using classical polynomials of degree $N$ inside each element, in our new approach the discrete solution is represented by piecewise continuous polynomials of degree $N$ within each Voronoi element, using a continuous finite element basis defined on a subgrid inside each polygon. We call the resulting subgrid basis an agglomerated finite element (AFE) basis for the DG method on general polygons, since it is obtained by the agglomeration of the finite element basis functions associated with the subgrid triangles. The basis functions on each sub-triangle are defined, as usual, on a universal reference element, hence allowing to compute universal mass, flux and stiffness matrices for the subgrid triangles once and for all in a pre-processing stage for the reference element only. Consequently, the construction of an efficient quadrature-free algorithm is possible, despite the unstructured nature of the computational grid. High order of accuracy in time is achieved thanks to the ADER approach, making use of an element-local space-time Galerkin finite element predictor.
The novel schemes are carefully validated against a set of typical benchmark problems for the compressible Euler and Navier-Stokes equations. The numerical results have been checked with reference solutions available in literature and also systematically compared, in terms of computational efficiency and accuracy, with those obtained by the corresponding modal DG version of the scheme.