Journal of Scientific Computing最新文献

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.

In this paper, a hybrid SBP-SAT/pseudo-spectral method is proposed for solving the time-dependent Wigner equation. High-order summation-by-parts (SBP) operators are utilized to discretize the Wigner equation spatially, where the inflow boundary conditions are weakly imposed by adding simultaneous approximation terms (SATs) to the semi-discretized Wigner equation. The pseudo-differential term, governing the quantum effect, is discretized in a pseudo-spectral manner with spectral accuracy. (L^2)-stabilities of both the semi-discretized (excluding time) and fully discretized systems are thoroughly discussed, with the inclusion of an arbitrary-stage explicit Runge–Kutta scheme for time integration. Numerical experiments are conducted, including simulations of a harmonic oscillator, a Gaussian wave packet, and a typical RTD with its I–V characteristic curves. The numerical results demonstrate: (1) the accuracy order of the numerical scheme in discretizing the Wigner equation in phase space matches the theoretical value; (2) observation of typical quantum effects, including tunneling and negative resistance; and (3) rapid convergence of numerical solutions relative to the accuracy order of SBP operators.

The optimal transport (OT) problem can be reduced to a linear programming (LP) problem through discretization. In this paper, we introduced the random block coordinate descent (RBCD) methods to directly solve this LP problem. Our approach involves restricting the potentially large-scale optimization problem to small LP subproblems constructed via randomly chosen working sets. By using a random Gauss-Southwell-q rule to select these working sets, we equip the vanilla version of (({textbf {RBCD}}_0)) with almost sure convergence and a linear convergence rate to solve general standard LP problems. To further improve the efficiency of the (({textbf {RBCD}}_0)) method, we explore the special structure of constraints in the OT problems and leverage the theory of linear systems to propose several approaches for refining the random working set selection and accelerating the vanilla method. Inexact versions of the RBCD methods are also discussed. Our preliminary numerical experiments demonstrate that the accelerated random block coordinate descent (ARBCD) method compares well with other solvers including stabilized Sinkhorn’s algorithm when seeking solutions with relatively high accuracy, and offers the advantage of saving memory.

Linearly implicit methods for Ordinary Differential Equations combined with the application of Approximate Matrix Factorization (AMF) provide efficient numerical methods for the solution of large semi-discrete parabolic Partial Differential Equations in several spatial dimensions. Interesting particular subclasses of such linearly implicit methods are the so-called W-methods and the TASE W-methods recently introduced in González-Pinto et al. (Appl Numer Math, 188:129–145, 2023) with the aim of reducing the computational cost of the TASE Runge–Kutta methods in Bassenne et al. (J Comput Phys 424:109847, 2021) and Calvo et al. (J Comp Phys 436:110316, 2021). In this paper, we study the application of the AMF approach in combination with TASE W-methods. While for AMF W-methods the temporal order of consistency is immediately obtained from that of the underlying W-method, this property needs a more thorough analysis for the newly introduced AMF-TASE W-methods. For these latter methods it is described which are the additional order conditions to be fulfilled and it is shown that the parallel structure of the methods is crucial to retain the order of consistency of the underlying TASE W-method. Numerical experiments are presented in three spatial dimensions to assess the consistency result and to show that the proposed schemes are competitive with other well-known good performing AMF W-methods.

This paper aims to develop an efficient numerical scheme for simulating solid-state dewetting with strongly anisotropic surface energies in two dimensions. The governing equation is a sixth-order, highly nonlinear geometric partial differential equation, which makes it quite challenging to design an efficient numerical scheme. To tackle this problem, we first introduce an appropriate tangent velocity of the interface curve which could help mesh points equally distribute along the curve, then we reformulate the governing equation in terms of the tangent angle (theta ) and the length L of the interface curve with a release of the stiffness brought by surface tension. To further reduce the numerical stability constraint from the high-order PDE, we propose a mixed finite element method for solving the reformulated (theta )-L equations. Numerical results are provided to demonstrate that the (theta )-L approach is not only efficient and accurate, but also has the mesh equidistribution property with an improved numerical stability.

We propose second-order numerical methods based on the generalized positive auxiliary variable (gPAV) framework for solving the Cahn–Hilliard–Navier–Stokes–Darcy model in superposed free flow and porous media. In the gPAV-reformulated system, we introduce an auxiliary variable according to the modified energy law and take account into the interface conditions between the two subdomains. By implicit-explicit temporal discretization, we develop fully decoupled linear gPAV-CNLF and gPAV-BDF2 numerical methods effected with the Galerkin finite element method. The fully discrete schemes satisfy a modified energy law irrespective of time step size. Plentiful numerical experiments are performed to validate the methods and demonstrate the robustness. The application in filtration systems, the influence of viscous instability, general permeability, curve interface, and different densities are discussed in details to further illustrate the compatibility and applicability of our developed gPAV numerical methods.

We propose the Compact Coupling Interface Method, a finite difference method capable of obtaining second-order accurate approximations of not only solution values but their gradients, for elliptic complex interface problems with interfacial jump conditions. Such elliptic interface boundary value problems with interfacial jump conditions are a critical part of numerous applications in fields such as heat conduction, fluid flow, materials science, and protein docking, to name a few. A typical example involves the construction of biomolecular shapes, where such elliptic interface problems are in the form of linearized Poisson–Boltzmann equations, involving discontinuous dielectric constants across the interface, that govern electrostatic contributions. Additionally, when interface dynamics are involved, the normal velocity of the interface might be comprised of the normal derivatives of solution, which can be approximated to second-order by our method, resulting in accurate interface dynamics. Our method, which can be formulated in arbitrary spatial dimensions, combines elements of the highly-regarded Coupling Interface Method, for such elliptic interface problems, and Smereka’s second-order accurate discrete delta function. The result is a variation and hybrid with a more compact stencil than that found in the Coupling Interface Method, and with advantages, borne out in numerical experiments involving both geometric model problems and complex biomolecular surfaces, in more robust error profiles.

In this paper, by introducing two temporal derivative-dependent auxiliary variables, a linearized and decoupled fourth-order compact finite difference method is developed and analyzed for the nonlinear coupled bacterial systems. The temporal-spatial error splitting technique and discrete energy method are employed to prove the unconditional stability and convergence of the method in discrete maximum-norm. Furthermore, to improve the computational efficiency, an alternating direction implicit (ADI) compact difference algorithm is proposed, and the unconditional stability and optimal-order maximum-norm error estimate for the ADI scheme are also strictly established. Finally, several numerical experiments are conducted to validate the theoretical convergence and to simulate the phenomena of bacterial extinction as well as the formation of endemic diseases. In particular, an adaptive time-stepping algorithm is developed and tested for long-term stable simulations.

We propose a novel methodology for forecasting spatio-temporal data using supervised semi-nonnegative matrix factorization (SSNMF) with frequency regularization. Matrix factorization is employed to decompose spatio-temporal data into spatial and temporal components. To improve clarity in the temporal patterns, we introduce a nonnegativity constraint on the time domain along with regularization in the frequency domain. Specifically, regularization in the frequency domain involves selecting features in the frequency space, making an interpretation in the frequency domain more convenient. We propose two methods in the frequency domain: soft and hard regularizations, and provide convergence guarantees to first-order stationary points of the corresponding constrained optimization problem. While our primary motivation stems from geophysical data analysis based on GRACE (Gravity Recovery and Climate Experiment) data, our methodology has the potential for wider application. Consequently, when applying our methodology to GRACE data, we find that the results with the proposed methodology are comparable to previous research in the field of geophysical sciences but offer clearer interpretability.