Numerical Methods

insight - Numerical Methods

Ensuring Numerical Reliability and Accuracy in Pipelined Bi-Conjugate Gradient Stabilized Method through ExBLAS Approach

The ExBLAS approach can provide reproducible and accurate results for the pipelined Bi-Conjugate Gradient Stabilized (p-BiCGStab) method, avoiding the need for residual replacement techniques.

Continuously Bounds-Preserving Discontinuous Galerkin Methods for Hyperbolic Conservation Laws

A novel limiting approach for discontinuous Galerkin methods is presented which ensures that the solution is continuously bounds-preserving for any arbitrary choice of basis, approximation order, and mesh element type.

High-Performance Matrix-Free Evaluation of Unfitted Finite Element Operators

This work presents efficient matrix-free algorithms for evaluating the operator action in unfitted finite element discretizations, enabling high-performance computations with high-order polynomial spaces.

Efficient Numerical Methods for Solving Fokker-Planck Equations with Positivity Preservation

Fokker-Planck equations can be efficiently solved using the Chang-Cooper method combined with unconditionally positive and conservative Patankar-type time integration schemes, which preserve positivity and steady states.

Efficient Multiscale Coarse Approximations for Diffusion Problems in Perforated Domains

The authors propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain to efficiently solve diffusion problems in perforated domains. The coarse space is spanned by locally discrete harmonic basis functions with piecewise polynomial traces along the subdomain boundaries. The method provides superconvergence for a specific edge refinement procedure, even if the true solution has low regularity.

A Finite Element Method for Evolving Multi-Phase Interfaces with Triple Junctions

The authors introduce a structure-preserving finite element method for the numerical approximation of the multi-phase Mullins-Sekerka problem, which models the evolution of a network of curves driven by surface energy minimization and subject to area preservation constraints.

Dynamical Low-Rank Approximation of the Vlasov-Poisson Equation with Piecewise Linear Spatial Boundary

The core message of this article is to develop a projector splitting scheme for dynamical low-rank approximation (DLRA) of the Vlasov-Poisson equation that can handle inflow boundary conditions on spatial domains with piecewise linear boundaries.

A Conservative Eulerian Finite Element Method for Scalar Transport and Diffusion in Moving Domains

The paper introduces a conservative Eulerian finite element method for the transport and diffusion of a scalar quantity in a time-dependent domain. The method is based on a reformulation of the partial differential equation to derive a scheme that conserves the quantity under consideration exactly on the discrete level.

Efficient Approximation of Parametric PDE Solutions using Radial Basis Functions, Proper Orthogonal Decomposition, and Deep Neural Networks

The authors propose a novel algorithm, POD-DNN, that leverages deep neural networks (DNNs) along with radial basis functions (RBFs) in the context of the proper orthogonal decomposition (POD) reduced basis method (RBM) to efficiently approximate the parametric mapping of parametric partial differential equations on irregular domains.

Tensor-Train Stochastic Finite Volume Method for Efficient Uncertainty Quantification of Hyperbolic Conservation Laws

The authors introduce a tensor-train reformulation of the stochastic finite volume (SFV) method to efficiently tackle high-dimensional uncertainty quantification problems for hyperbolic conservation laws involving shocks and discontinuities.

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