Despite the growing interest in parallel-in-time methods as an approach to accelerate numerical simulations in atmospheric modelling, improving their stability and convergence remains a substantial challenge for their application to operational models. In this work, we study the temporal parallelization of the shallow water equations on the rotating sphere combined with time-stepping schemes commonly used in atmospheric modelling due to their stability properties, namely an Eulerian implicit-explicit (IMEX) method and a semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to investigate the performance of parallel-in-time methods, namely Parareal and Multigrid Reduction in Time (MGRIT), when these well-established schemes are used on the coarse discretization levels and provide insights on how they can be improved for better performance. We begin by performing an analytical stability study of Parareal and MGRIT applied to a linearized ordinary differential equation depending on the choice of a coarse scheme. Next, we perform numerical simulations of two standard tests to evaluate the stability, convergence and speedup provided by the parallel-in-time methods compared to a fine reference solution computed serially. We also conduct a detailed investigation on the influence of artificial viscosity and hyperviscosity approaches, applied on the coarse discretization levels, on the performance of the temporal parallelization. Both the analytical stability study and the numerical simulations indicate a poorer stability behaviour when SL-SI-SETTLS is used on the coarse levels, compared to the IMEX scheme. With the IMEX scheme, a better trade-off between convergence, stability and speedup compared to serial simulations can be obtained under proper parameters and artificial viscosity choices, opening the perspective of the potential competitiveness for realistic models.
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We propose a new method to construct a stationary process and random field with a given convex, decreasing covariance function and any one-dimensional marginal distribution. The result is a new class of stationary processes and random fields. The construction method provides a simple, unified approach for a wide range of covariance functions and any one-dimensional marginal distributions, and it allows a new way to model dependence structures in a stationary process/random field as its dependence structure is induced by the correlation structure of a few disjoint sets in the support set of the marginal distribution.
This work is concerned with solving high-dimensional Fokker-Planck equations with the novel perspective that solving the PDE can be reduced to independent instances of density estimation tasks based on the trajectories sampled from its associated particle dynamics. With this approach, one sidesteps error accumulation occurring from integrating the PDE dynamics on a parameterized function class. This approach significantly simplifies deployment, as one is free of the challenges of implementing loss terms based on the differential equation. In particular, we introduce a novel class of high-dimensional functions called the functional hierarchical tensor (FHT). The FHT ansatz leverages a hierarchical low-rank structure, offering the advantage of linearly scalable runtime and memory complexity relative to the dimension count. We introduce a sketching-based technique that performs density estimation over particles simulated from the particle dynamics associated with the equation, thereby obtaining a representation of the Fokker-Planck solution in terms of our ansatz. We apply the proposed approach successfully to three challenging time-dependent Ginzburg-Landau models with hundreds of variables.
Integro-differential equations, analyzed in this work, comprise an important class of models of continuum media with nonlocal interactions. Examples include peridynamics, population and opinion dynamics, the spread of disease models, and nonlocal diffusion, to name a few. They also arise naturally as a continuum limit of interacting dynamical systems on networks. Many real-world networks, including neuronal, epidemiological, and information networks, exhibit self-similarity, which translates into self-similarity of the spatial domain of the continuum limit. For a class of evolution equations with nonlocal interactions on self-similar domains, we construct a discontinuous Galerkin method and develop a framework for studying its convergence. Specifically, for the model at hand, we identify a natural scale of function spaces, which respects self-similarity of the spatial domain, and estimate the rate of convergence under minimal assumptions on the regularity of the interaction kernel. The analytical results are illustrated by numerical experiments on a model problem.
We propose a new space-time variational formulation for wave equation initial-boundary value problems. The key property is that the formulation is coercive (sign-definite) and continuous in a norm stronger than $H^1(Q)$, $Q$ being the space-time cylinder. Coercivity holds for constant-coefficient impedance cavity problems posed in star-shaped domains, and for a class of impedance-Dirichlet problems. The formulation is defined using simple Morawetz multipliers and its coercivity is proved with elementary analytical tools, following earlier work on the Helmholtz equation. The formulation can be stably discretised with any $H^2(Q)$-conforming discrete space, leading to quasi-optimal space-time Galerkin schemes. Several numerical experiments show the excellent properties of the method.
Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations { in the $L^\infty(0, T; L^2(\Omega; L^2))$ norm} all suffer from the order reduction with respect to the spatial discretizations. The best convergence result obtained for these fully discrete schemes is only half-order in time and first-order in space, which is not optimal in space in the traditional sense. The objective of this article is to establish strong convergence of $O(\tau^{1/2}+ h^2)$ in the $L^\infty(0, T; L^2(\Omega; L^2))$ norm for approximating the velocity, and strong convergence of $O(\tau^{1/2}+ h)$ in the $L^{\infty}(0, T;L^2(\Omega;L^2))$ norm for approximating the time integral of pressure, where $\tau$ and $h$ denote the temporal step size and spatial mesh size, respectively. The error estimates are of optimal order for the spatial discretization considered in this article (with MINI element), and consistent with the numerical experiments. The analysis is based on the fully discrete Stokes semigroup technique and the corresponding new estimates.
This paper introduces a time-domain combined field integral equation for electromagnetic scattering by a perfect electric conductor. The new equation is obtained by leveraging the quasi-Helmholtz projectors, which separate both the unknown and the source fields into solenoidal and irrotational components. These two components are then appropriately rescaled to cure the solution from a loss of accuracy occurring when the time step is large. Yukawa-type integral operators of a purely imaginary wave number are also used as a Calderon preconditioner to eliminate the ill-conditioning of matrix systems. The stabilized time-domain electric and magnetic field integral equations are linearly combined in a Calderon-like fashion, then temporally discretized using a proper pair of trial functions, resulting in a marching-on-in-time linear system. The novel formulation is immune to spurious resonances, dense discretization breakdown, large-time step breakdown and dc instabilities stemming from non-trivial kernels. Numerical results for both simply-connected and multiply-connected scatterers corroborate the theoretical analysis.
In this work we consider the two dimensional instationary Navier-Stokes equations with homogeneous Dirichlet/no-slip boundary conditions. We show error estimates for the fully discrete problem, where a discontinuous Galerkin method in time and inf-sup stable finite elements in space are used. Recently, best approximation type error estimates for the Stokes problem in the $L^\infty(I;L^2(\Omega))$, $L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms have been shown. The main result of the present work extends the error estimate in the $L^\infty(I;L^2(\Omega))$ norm to the Navier-Stokes equations, by pursuing an error splitting approach and an appropriate duality argument. In order to discuss the stability of solutions to the discrete primal and dual equations, a specially tailored discrete Gronwall lemma is presented. The techniques developed towards showing the $L^\infty(I;L^2(\Omega))$ error estimate, also allow us to show best approximation type error estimates in the $L^2(I;H^1(\Omega))$ and $L^2(I;L^2(\Omega))$ norms, which complement this work.
Maxwell-Amp\`{e}re-Nernst-Planck (MANP) equations were recently proposed to model the dynamics of charged particles. In this study, we enhance a numerical algorithm of this system with deep learning tools. The proposed hybrid algorithm provides an automated means to determine a proper approximation for the dummy variables, which can otherwise only be obtained through massive numerical tests. In addition, the original method is validated for 2-dimensional problems. However, when the spatial dimension is one, the original curl-free relaxation component is inapplicable, and the approximation formula for dummy variables, which works well in a 2-dimensional scenario, fails to provide a reasonable output in the 1-dimensional case. The proposed method can be readily generalised to cases with one spatial dimension. Experiments show numerical stability and good convergence to the steady-state solution obtained from Poisson-Boltzmann type equations in the 1-dimensional case. The experiments conducted in the 2-dimensional case indicate that the proposed method preserves the conservation properties.
Finite-dimensional truncations are routinely used to approximate partial differential equations (PDEs), either to obtain numerical solutions or to derive reduced-order models. The resulting discretized equations are known to violate certain physical properties of the system. In particular, first integrals of the PDE may not remain invariant after discretization. Here, we use the method of reduced-order nonlinear solutions (RONS) to ensure that the conserved quantities of the PDE survive its finite-dimensional truncation. In particular, we develop two methods: Galerkin RONS and finite volume RONS. Galerkin RONS ensures the conservation of first integrals in Galerkin-type truncations, whether used for direct numerical simulations or reduced-order modeling. Similarly, finite volume RONS conserves any number of first integrals of the system, including its total energy, after finite volume discretization. Both methods are applicable to general time-dependent PDEs and can be easily incorporated in existing Galerkin-type or finite volume code. We demonstrate the efficacy of our methods on two examples: direct numerical simulations of the shallow water equation and a reduced-order model of the nonlinear Schrodinger equation. As a byproduct, we also generalize RONS to phenomena described by a system of PDEs.
We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value $1$ at $(1,1,\ldots, 1)$. We characterize the existence of a $k$-increasing $n$-variate function $C$ fulfilling $A\leq C\leq B$ for standardized $n$-variate functions $A,B$ and discuss the method for constructing this function. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when $A$ respectively $B$ coincides with the pointwise infimum respectively supremum of the set of all $k$-increasing $n$-variate functions $C$ fulfilling $A\leq C\leq B$.
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