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This paper proposes and analyzes a novel efficient high-order finite volume method for the ideal magnetohydrodynamics (MHD). As a distinctive feature, the method simultaneously preserves two critical physical constraints: a discretely divergence-free (DDF) constraint on the magnetic field and the positivity-preserving (PP) property, which ensures the positivity of density, pressure, and internal energy. To enforce the DDF condition in each cell, we design a new discrete projection approach that projects the reconstructed point values at the cell interface into a DDF space, without using any approximation polynomials. This projection method is highly efficient, easy to implement, and particularly suitable for the high-order finite volume methods that return only the point values (no explicit approximation polynomials) in the reconstruction. Moreover, we also develop a new finite volume framework for constructing provably PP schemes for the ideal MHD system. The framework comprises the discrete projection technique, a suitable approximation to the Godunov--Powell source terms, and a simple PP limiter. We provide rigorous analysis of the PP property of the proposed finite volume method, demonstrating that the DDF condition and the proper approximation to the source terms eliminate the impact of magnetic divergence terms on the PP property. The analysis is challenging due to the internal energy function's nonlinearity and the intricate relationship between the DDF and PP properties. To address these challenges, we adopt the recently developed geometric quasilinearization approach [K. Wu and C.-W. Shu, SIAM Rev., 65 (2023), pp. 1031--1073], which transforms a nonlinear constraint into a family of linear constraints. Finally, we validate the effectiveness of the proposed method through several benchmark and demanding numerical examples. The results demonstrate that the proposed method is robust, accurate, and highly effective, confirming the significance of the proposed DDF projection and PP techniques. [ABSTRACT FROM AUTHOR]

In this paper we consider a numerical homogenization technique for curl-curl problems that is based on the framework of the localized orthogonal decomposition and which was proposed in [D. Gallistl, P. Henning, and B. Verfürth, SIAM J. Numer. Anal., 56 (2018), pp. 1570-1596] for problems with essential boundary conditions. The findings of the aforementioned work establish quantitative homogenization results for the time-harmonic Maxwell's equations that hold beyond assumptions of periodicity; however, a practical realization of the approach was left open. In this paper, we transfer the findings from essential boundary conditions to natural boundary conditions, and we demonstrate that the approach yields a computable numerical method. We also investigate how boundary values of the source term can affect the computational complexity and accuracy. Our findings will be supported by various numerical experiments, both in 2D and 3D. [ABSTRACT FROM AUTHOR]

In this paper, we present a new regularization scheme for the linearized Poisson{ Boltzmann equation (PBE) which models the electrostatic potential of biomolecules in a solvent. This scheme is based on the splitting of the target potential into the short- and long-range components localized in the molecular region by using the range-separated (RS) tensor format [P. Benner, V. Khoromskaia, and B. N. Khoromskij, SIAM J. Comput., 2 (2018), pp. A1034{A1062] for representation of the discretized multiparticle Dirac delta [B. N. Khoromskij, J. Comput. Phys., 401 (2020), 108998] constituting the highly singular right-hand side in the PBE. From the computational point of view our regularization approach requires only the modification of the right-hand side in the PBE so that it can be implemented within any open-source grid-based software package for solving PBE that already includes some FEM/FDM disretization scheme for elliptic PDE and solver for the arising linear system of equations. The main computational benefits are twofold. First, one applies the chosen PBE solver only for the smooth long-range (regularized) part of the collective potential with the regular right-hand side represented by a low-rank RS tensor with a controllable precision. Thus, we eliminate the numerical treatment of the singularities in the right-hand side and do not change the interface and boundary conditions. And second, the elliptic PDE need not be solved for the singular part in the right-hand side at all, since the short-range part of the target potential of the biomolecule is precomputed independently on a computational grid by simple one-dimensional tensor operations. The total potential is then obtained by adding the numerical solution of the PBE for the smooth long-range part to the directly precomputed tensor representation for the short-range contribution. Numerical tests illustrate that the new regularization scheme, implemented by a simple modification of the right-hand side in the chosen PBE solver, improves the accuracy of the approximate solution on rather coarse grids. The scheme also demonstrates good convergence behavior on a sequence of refined grids. [ABSTRACT FROM AUTHOR]

We analyze the recently introduced family of preconditioners in [M. M. Rana et al., SIAM J. Sci. Comput., 43 (2021), pp. S475-S495] for the stage equations of implicit Runge--Kutta methods for s-stage methods. We simplify the formulas for the eigenvalues and eigenvectors of the preconditioned systems for a general s-stage method and use these to obtain convergence rate estimates for preconditioned GMRES for some common choices of the implicit Runge-Kutta methods. This analysis is based on understanding the inherent matrix structure of these problems and exploiting it to qualitatively predict and explain the main observed features of the GMRES convergence behavior, using tools from approximation and potential theory based on Schwarz--Christoffel maps for curves and close, connected domains in the complex plane. We illustrate our analysis with numerical experiments showing very close correspondence of the estimates and the observed behavior, suggesting the analysis reliably captures the essence of these preconditioners. [ABSTRACT FROM AUTHOR]

In [L. Li, M. Ni, and W. Zheng, SIAM J. Sci. Comput., 41 (2019), pp. B796-B815] a charge-conservative finite element method is proposed for solving inductionless and incompressible magnetohydrodynamic (MHD) equations. The purpose of this paper is to propose a robust solver for the discrete problem. Using the framework of field-of-values-equivalence, we first study the preconditioned Krylov space method for the continuous problem in the setting of Hilbert spaces. The algebraic preconditioner for the discrete problem is then obtained by representing the preconditioner for the continuous problem in finite element spaces. By three numerical examples, the optimality of the solver to the number of unknowns is demonstrated for both stationary and time-dependent MHD problems. [ABSTRACT FROM AUTHOR]

The study of collision-induced nonlinear breakage phenomenon is mostly unexplored but is important in the area of particulate processes. In this work, the volume and time dependent collisional breakage kernel function is modeled based on the population balance modeling approach. To solve the nonlinear breakage population balance equation, the weighted finite volume scheme for linear breakage process from Kumar, Saha, and Tsotsas [SIAM J. Numer. Anal., 53 (2015), pp. 1672--1689] is extended for the case of collision-induced breakage process. The weighted finite volume scheme is developed in such a way that it conserves the total mass of the system while preserving the total number of particles in the system. Moreover, an event-driven constant number Monte Carlo simulation algorithm is presented, and the simulation results are used as an alternative to experimental results. The volume dependency of the collisional breakage kernel is incorporated successfully in the Monte Carlo simulation for the first time while selecting particles for collision events. Some essential properties of any particulate process, such as the total number of particles and the size distribution of particles, are validated successfully for several breakage distribution functions using the Monte Carlo results. This offers new insights into the estimation and interpretation of collision-induced nonlinear breakage kinetics. [ABSTRACT FROM AUTHOR]