We study a first-order system formulation of the (acoustic) wave equation and prove that the operator of this system is an isomorphsim from an appropriately defined graph space to L^2. The results rely on well-posedness and stability of the weak and ultraweak formulation of the second-order wave equation. As an application we define and analyze a space-time least-squares finite element method for solving the wave equation. Some numerical examples for one- and two- dimensional spatial domains are presented.
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We consider the parallel-in-time solution of scalar nonlinear conservation laws in one spatial dimension. The equations are discretized in space with a conservative finite-volume method using weighted essentially non-oscillatory (WENO) reconstructions, and in time with high-order explicit Runge-Kutta methods. The solution of the global, discretized space-time problem is sought via a nonlinear iteration that uses a novel linearization strategy in cases of non-differentiable equations. Under certain choices of discretization and algorithmic parameters, the nonlinear iteration coincides with Newton's method, although, more generally, it is a preconditioned residual correction scheme. At each nonlinear iteration, the linearized problem takes the form of a certain discretization of a linear conservation law over the space-time domain in question. An approximate parallel-in-time solution of the linearized problem is computed with a single multigrid reduction-in-time (MGRIT) iteration. The MGRIT iteration employs a novel coarse-grid operator that is a modified conservative semi-Lagrangian discretization and generalizes those we have developed previously for non-conservative scalar linear hyperbolic problems. Numerical tests are performed for the inviscid Burgers and Buckley--Leverett equations. For many test problems, the solver converges in just a handful of iterations with convergence rate independent of mesh resolution, including problems with (interacting) shocks and rarefactions.
We consider the dynamic Biot model describing the interaction between fluid flow and solid deformation including wave propagation phenomena in both the liquid and solid phases of a saturated porous medium. The model couples a hyperbolic equation for momentum balance to a second-order in time dynamic Darcy law and a parabolic equation for the balance of mass and is here considered in three-field formulation with the displacement of the elastic matrix, the fluid velocity, and the fluid pressure being the physical fields of interest. A family of variational space-time finite element methods is proposed that combines a continuous-in-time Galerkin ansatz of arbitrary polynomial degree with inf-sup stable $H(\rm{div})$-conforming approximations of discontinuous Galerkin (DG) type in case of the displacement and a mixed approximation of the flux, its time derivative and the pressure field. We prove error estimates in a combined energy norm as well as $L^2$~error estimates in space for the individual fields for both maximum and $L^2$ norm in time which are optimal for the displacement and pressure approximations.
We propose a method to numerically compute fractional derivatives (or the fractional Laplacian) on the whole real line via Riesz fractional integrals. The compactified real line is divided into a number of intervals, thus amounting to a multi-domain approach; after transformations in accordance with the underlying $Z_{q}$ curve ensuring analyticity of the respective integrands, the integrals over the different domains are computed with a Clenshaw-Curtis algorithm. As an example, we consider solitary waves for fractional Korteweg-de Vries equations and compare these to results obtained with a discrete Fourier transform.
The present work provides a comprehensive study of symmetric-conjugate operator splitting methods in the context of linear parabolic problems and demonstrates their additional benefits compared to symmetric splitting methods. Relevant applications include nonreversible systems and ground state computations for linear Schr\"odinger equations based on the imaginary time propagation. Numerical examples confirm the favourable error behaviour of higher-order symmetric-conjugate splitting methods and illustrate the usefulness of a time stepsize control, where the local error estimation relies on the computation of the imaginary parts and thus requires negligible costs.
The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices $A_n$ arising from numerical discretizations of differential equations. Indeed, when the mesh fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to a sequence $\{A_n\}_n$, which often turns out to be a GLT sequence. In this paper, we extend the theory of GLT sequences in several directions: we show that every GLT sequence enjoys a normal form, we identify the spectral symbol of every GLT sequence formed by normal matrices, and we prove that, for every GLT sequence $\{A_n\}_n$ formed by normal matrices and every continuous function $f:\mathbb C\to\mathbb C$, the sequence $\{f(A_n)\}_n$ is again a GLT sequence whose spectral symbol is $f(\kappa)$, where $\kappa$ is the spectral symbol of $\{A_n\}_n$. In addition, using the theory of GLT sequences, we prove a spectral distribution result for perturbed normal matrices.
The measurement of deep water gravity wave elevations using in-situ devices, such as wave gauges, typically yields spatially sparse data. This sparsity arises from the deployment of a limited number of gauges due to their installation effort and high operational costs. The reconstruction of the spatio-temporal extent of surface elevation poses an ill-posed data assimilation problem, challenging to solve with conventional numerical techniques. To address this issue, we propose the application of a physics-informed neural network (PINN), aiming to reconstruct physically consistent wave fields between two designated measurement locations several meters apart. Our method ensures this physical consistency by integrating residuals of the hydrodynamic nonlinear Schr\"{o}dinger equation (NLSE) into the PINN's loss function. Using synthetic wave elevation time series from distinct locations within a wave tank, we initially achieve successful reconstruction quality by employing constant, predetermined NLSE coefficients. However, the reconstruction quality is further improved by introducing NLSE coefficients as additional identifiable variables during PINN training. The results not only showcase a technically relevant application of the PINN method but also represent a pioneering step towards improving the initialization of deterministic wave prediction methods.
It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions required. It becomes crucial to properly choose effective basis functions, such that the dimension of the stochastic approximation space can be reduced. In this work, we focus on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC), and explore the gPC expansion based on the analysis of variance (ANOVA) decomposition. A concise form of the gPC expansion is presented for each component function of the ANOVA expansion, and an adaptive ANOVA procedure is proposed to construct the overall stochastic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA stochastic Galerkin method for both diffusion and Helmholtz problems.
We consider the surface Stokes equation on a smooth closed hypersurface in three-dimensional space. For discretization of this problem a generalization of the surface finite element method (SFEM) of Dziuk-Elliott combined with a Hood-Taylor pair of finite element spaces has been used in the literature. We call this method Hood-Taylor-SFEM. This method uses a penalty technique to weakly satisfy the tangentiality constraint. In this paper we present a discretization error analysis of this method resulting in optimal discretization error bounds in an energy norm. We also address linear algebra aspects related to (pre)conditioning of the system matrix.
This study develops a computationally efficient phase-field lattice Boltzmann model with the capability to simulate thermocapillary flows. The model was implemented into the open-source simulation framework, waLBerla, and extended to conduct the collision stage using central moments. The multiphase model was coupled with both a passive-scalar thermal LB, and a RK solution to the energy equation in order to resolve temperature-dependent surface tension phenomena. Various lattice stencils (D3Q7, D3Q15, D3Q19, D3Q27) were tested for the passive-scalar LB and both the second- and fourth-order RK methods were investigated. There was no significant difference observed in the accuracy of the LB or RK schemes. The passive scalar D3Q7 LB discretisation tended to provide computational benefits, while the second order RK scheme is superior in memory usage. This paper makes contributions relating to the modelling of thermocapillary flows and to understanding the behaviour of droplet capture with thermal sources analogous to thermal tweezers. Four primary contributions to the literature are identified. First, a new 3D thermocapillary, central-moment phase-field LB model is presented and implemented in the open-source software, waLBerla. Second, the accuracy and computational performance of various techniques to resolve the energy equation for multiphase, incompressible fluids is investigated. Third, the dynamic droplet transport behaviour in the presence of thermal sources is studied and insight is provided on the potential ability to manipulate droplets based on local domain heating. Finally, a concise analysis of the computational performance together with near-perfect scaling results on NVIDIA and AMD GPU-clusters is shown. This research enables the detailed study of droplet manipulation and control in thermocapillary devices.
This work deals with an inverse source problem for the biharmonic wave equation. A two-stage numerical method is proposed to identify the unknown source from the multi-frequency phaseless data. In the first stage, we introduce some artificially auxiliary point sources to the inverse source system and establish a phase retrieval formula. Theoretically, we point out that the phase can be uniquely determined and estimate the stability of this phase retrieval approach. Once the phase information is retrieved, the Fourier method is adopted to reconstruct the source function from the phased multi-frequency data. The proposed method is easy-to-implement and there is no forward solver involved in the reconstruction. Numerical experiments are conducted to verify the performance of the proposed method.
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