In this work, we establish that discontinuous Galerkin methods are capable of producing reliable approximations for a broad class of nonlinear variational problems. In particular, we demonstrate that these schemes pro...
The finite volume, self-adaptive theta (SATh) scheme was defined in Arbogast and Huang, A self-adaptive theta scheme using discontinuity aware quadrature for solving conservation laws, IMA J. Numer. Anal. (2022). The ...
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The finite volume, self-adaptive theta (SATh) scheme was defined in Arbogast and Huang, A self-adaptive theta scheme using discontinuity aware quadrature for solving conservation laws, IMA J. Numer. Anal. (2022). The basic scheme evolves both the local space and space-time averages of the solution in time with an implicitly defined theta parameter. Here, the scheme is extended to unstructured meshes in multiple space dimensions, general numerical flux functions, and higher (formally second) order using WENO reconstructions. Theoretical results apply to the one space dimension, upstream weighted case, in the setting of a monotone solution. In this case, if the theta parameter is bounded below by $$\theta _{\min }=0$$ , it is shown that SATh is stable, L-stable for the linear problem, total variation diminishing (TVD), and maximum principle preserving (MPP). These results generalize those known previously with the assumption that $$\theta _{\min }=1/2$$ . Numerical tests for problems with contact discontinuities, shocks, and rarefactions show that SATh performs better than finite volume schemes using backward Euler time stepping. Moreover, SATh gives solutions about as sharp as when using Crank-Nicolson time stepping, but SATh is non-oscillatory. In cases covered by the theoretical results, SATh combined with a Lax-Friedrichs numerical flux (rather than upstream weighting) appears to be TVD and MPP. SATh is non-oscillatory if $$\theta _{\min }=1/2$$ , but if $$\theta _{\min }=0$$ and the solution is not monotone, it can develop oscillations. The higher order SATh scheme converges to order two and compares favorably with CN, but is less oscillatory.
The Keller-Segel equation, a classical chemotaxis model, and many of its variants have been extensively studied for decades. In this work, we focus on 3D Keller-Segel equation with a quadratic logistic damping term −&...
This paper presents a multilevel tensor compression algorithm called tensor butterfly algorithm for efficiently representing large-scale and high-dimensional oscillatory integral operators, including Green’s function...
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This paper presents a multilevel tensor compression algorithm called tensor butterfly algorithm for efficiently representing large-scale and high-dimensional oscillatory integral operators, including Green’s functions for wave equations and integral transforms such as Radon transforms and Fourier transforms. The proposed algorithm leverages a tensor extension of the so-called complementary low-rank property of existing matrix butterfly algorithms. The algorithm partitions the discretized integral operator tensor into subtensors of multiple levels and factorizes each subtensor at the middle level as a Tucker-type interpolative decomposition, whose factor matrices are formed in a multilevel fashion. For a -dimensional () integral operator discretized into a
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We introduce the Riemannian Proximal Sampler, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the Manifold Brownian Incremen...
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We report a molecular dynamics study of ab initio quality of the ferroelectric phase transition in crystalline PbTiO3. We model anharmonicity accurately in terms of potential energy and polarization surfaces trained o...
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We report a molecular dynamics study of ab initio quality of the ferroelectric phase transition in crystalline PbTiO3. We model anharmonicity accurately in terms of potential energy and polarization surfaces trained on density functional theory data with modern machine learning techniques. Our simulations demonstrate that the transition has a strong order-disorder character, in agreement with diffraction experiments, and provide fresh insight into the approach to equilibrium across the phase transition. We find that the emergence and disappearance of the macroscopic polarization is driven by dipolar switching at the nanometer scale. We also computed the infrared optical absorption spectra in both the ferroelectric and the paraelectric phases, finding good agreement with the experimental Raman frequencies. Often, the almost ideal displacive character of the soft mode detected by Raman scattering in the paraelectric phase has been contrasted with the order-disorder character of the transition suggested by diffraction experiments. We settle this issue by showing that the soft mode coexists with a strong Debye relaxation associated with thermal disordering of the dipoles. The Debye relaxation feature is centered at zero frequency and appears near the transition temperature in both the ferroelectric and the paraelectric phases.
Entropy stable discontinuous Galerkin (DG) methods improve the robustness of high order DG simulations of nonlinear conservation laws. These methods yield a semi-discrete entropy inequality, and rely on an algebraic f...
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We derive global estimates for the error in solutions of linear hyperbolic systems due to inaccurate boundary geometry. We show that the error is bounded by data and bounded in time when the solutions in the true and ...
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We propose P-order (Power-order), a unified, norm-independent framework for quantifying the convergence rates of iterative methods. Standard analyses based on Q-order are norm-dependent and require some uniformity of ...
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