Journal of Scientific Computing最新文献

We propose a robust randomized indicator method for the reliable detection of eigenvalue existence within an interval for symmetric matrices A. An indicator tells the eigenvalue existence based on some statistical norm estimators for a spectral projector. Previous work on eigenvalue indicators relies on a threshold which is empirically chosen, thus often resulting in under or over detection. In this paper, we use rigorous statistical analysis to guide the design of a robust indicator. Multiple randomized estimators for a contour integral operator in terms of A are analyzed. In particular, when A has eigenvalues inside a given interval, we show that the failure probability (for the estimators to return very small estimates) is extremely low. This enables to design a robust rejection indicator based on the control of the failure probability. We also give a prototype framework to illustrate how the indicator method may be applied numerically for eigenvalue detection and may potentially serve as a new way to design randomized symmetric eigenvalue solvers. Unlike previous indicator methods that only detect eigenvalue existence, the framework also provides a way to find eigenvectors with little extra cost by reusing computations from indicator evaluations. Extensive numerical tests show the reliability of the eigenvalue detection in multiple aspects.

We construct and analyze a multiscale finite element method for an elliptic distributed optimal control problem with pointwise control constraints, where the state equation has rough coefficients. We show that the performance of the multiscale finite element method is similar to the performance of standard finite element methods for smooth problems and present corroborating numerical results.

In this work, the immersed boundary method with time filter and scalar auxiliary variable techniques is studied to solve the fluid–structure interaction problems. For the fluid flow, we first use the backward Euler differentiation formula in temporal discretization, we then employ the time filter technique to improve its convergence order, the scalar auxiliary variable strategy is visited to decouple the fluid equations and achieve fast solutions. We adopt the immersed boundary method to build the connection between the fluid and the structure, as well as characterize the deformations of the structure. We approximate the fluid–structure interaction model by the finite element method in space. The semi-discrete and fully-discrete implicit numerical schemes are proposed followed with unconditionally stability properties. We carry out several numerical experiments to validate the convergence behaviors and efficiency of the algorithms.

We introduce two hybridizable discontinuous Galerkin (HDG) methods for numerically solving the two-dimensional Monge–Ampère equation. The first HDG method is devised to solve the nonlinear elliptic Monge–Ampère equation by using Newton’s method. The second HDG method is devised to solve a sequence of the Poisson equation until convergence to a fixed-point solution of the Monge–Ampère equation is reached. Numerical examples are presented to demonstrate the convergence and accuracy of the HDG methods. Furthermore, the HDG methods are applied to r-adaptive mesh generation by redistributing a given scalar density function via the optimal transport theory. This r-adaptivity methodology leads to the Monge–Ampère equation with a nonlinear Neumann boundary condition arising from the optimal transport of the density function to conform the resulting high-order mesh to the boundary. Hence, we extend the HDG methods to treat the nonlinear Neumann boundary condition. Numerical experiments are presented to illustrate the generation of r-adaptive high-order meshes on planar and curved domains.

The paper focuses on first-order invariant-domain preserving approximations of hyperbolic systems. We propose a new way to estimate the artificial viscosity that has to be added to make explicit, conservative, consistent numerical methods invariant-domain preserving and entropy inequality compliant. Instead of computing an upper bound on the maximum wave speed in Riemann problems, we estimate a minimum wave speed in the said Riemann problems such that the approximation satisfies predefined invariant-domain properties and predefined entropy inequalities. This technique eliminates non-essential fast waves from the construction of the artificial viscosity, while preserving pre-assigned invariant-domain properties and entropy inequalities.

An area-preserving parameterization of a surface is a bijective mapping that maps the surface onto a specified domain and preserves the local area. This paper formulates the computation of disk area-preserving parameterization as an improved optimization problem and develops a preconditioned nonlinear conjugate gradient method with guaranteed theoretical convergence for solving the problem. Numerical experiments indicate that our new approach has significantly improved area-preserving accuracy and computational efficiency compared to other state-of-the-art algorithms. Furthermore, we present an application of surface registration to illustrate the practical utility of area-preserving mappings as parameterizations of surfaces.

A novel approach for the stabilization of the Discontinuous Galerkin method based on the Dafermos entropy rate crition is presented. First, estimates for the maximal possible entropy dissipation rate of a weak solution are derived. Second, families of conservative Hilbert–Schmidt operators are identified to dissipate entropy. Steering these operators using the bounds on the entropy dissipation results in high-order accurate shock-capturing DG schemes for the one-dimensional Euler equations, satisfying the entropy rate criterion and an entropy inequality. Other testcases include the one-dimensional Buckley–Leverett equation.

Variable-exponent weakly singular Volterra integral equations of the second kind with integral kernels of the form ((t-s)^{-alpha (t)}) are considered. In Liang and Stynes (IMA J Numer Anal 19:drad072, 2023) it is shown that a typical solution of such an equation exhibits a weak singularity at the initial time (t=0), similarly to the case where (alpha (t)) is constant. Our paper extends this analysis further by giving a decomposition for the exact solution. To solve the problem numerically, a fractional polynomial collocation method is applied on a graded mesh. The convergence of the collocation solution to the exact solution is analysed rigorously and it is proved that specific choices of the fractional polynomials and mesh grading yield optimal-order convergence of the computed solution. Superconvergence properties of the iterated collocation solution are also analysed. Numerical experiments illustrate the sharpness of our theoretical results.

Excessive spatial parallelization can introduce a performance bottleneck due to the communication overhead. While time-parallel method multigrid-reduction-in-time (MGRIT) provides an alternative to enhance concurrency, it generally requires large numbers of iterations to converge or even fails when applied to advection-dominated problems. To enhance the convergence of MGRIT, we propose the use of consecutive-step coarse-grid operators in MGRIT, rather than the standard rediscretized coarse-grid operators. The consecutive-step coarse-grid operator is defined as the multiplication of several fine-grid operators, which is able to track the advective characteristic more accurately than the standard rediscretized one. Numerical results show that multilevel MGRIT using the proposed operator is more efficient than the one using the standard rediscretized operator when applied to linear advection problems. Moreover, we perform time-parallel computing of the Euler equations and the Navier–Stokes equations by using the proposed method. Spatial coarsening is also considered. Compared with the MGRIT using the standard rediscretization approach, the developed method demonstrates enhanced robustness and efficiency in handling complex flow problems, including cases involving multidimensional shock waves and contact discontinuities.

Many neurodegenerative diseases are connected to the spreading of misfolded prionic proteins. In this paper, we analyse the process of misfolding and spreading of both (alpha )-synuclein and Amyloid-(beta ), related to Parkinson’s and Alzheimer’s diseases, respectively. We introduce and analyze a positivity-preserving numerical method for the discretization of the Fisher-Kolmogorov equation, modelling accumulation and spreading of prionic proteins. The proposed approximation method is based on the discontinuous Galerkin method on polygonal and polyhedral grids for space discretization and on (vartheta -)method time integration scheme. We prove the existence of the discrete solution and a convergence result where the Implicit Euler scheme is employed for time integration. We show that the proposed approach is structure-preserving, in the sense that it guarantees that the discrete solution is non-negative, a feature that is of paramount importance in practical application. The numerical verification of our numerical model is performed both using a manufactured solution and considering wavefront propagation in two-dimensional polygonal grids. Next, we present a simulation of (alpha )-synuclein spreading in a two-dimensional brain slice in the sagittal plane. The polygonal mesh for this simulation is agglomerated maintaining the distinction of white and grey matter, taking advantage of the flexibility of PolyDG methods in the mesh construction. Finally, we simulate the spreading of Amyloid-(beta ) in a patient-specific setting by using a three-dimensional geometry reconstructed from magnetic resonance images and an initial condition reconstructed from positron emission tomography. Our numerical simulations confirm that the proposed method is able to capture the evolution of Parkinson’s and Alzheimer’s diseases.