数值分析
The least-squares neural network (LSNN) method introduced in [5] for linear advection-reaction equations is capable of accurately approximating discontinuous solutions without a priori knowledge of the interface location. However, the…
Implicit methods are a natural approach for the integration of stiff differential equations, to avoid time-step restrictions faced by standard explicit integrators. Explicit stabilised integrators are an alternative to implicit methods,…
Recently we introduced the least-squares algebraic-multigrid domain-decomposition (LS-AMG-DD) method as a multilevel, algebraic preconditioner for sparse symmetric positive definite (SPD) matrices that admit a Gram representation…
This paper presents a construction of a projector from an infinite-dimensional Hilbert complex of differential $k$-forms onto a finite-dimensional piecewise polynomial sub-complex. We demonstrate that, on contractable domains, the proposed…
In parametrized linear systems $\mathsf{P}(\mu)\mathsf{x}=\mathsf{b}$ the system matrix $\mathsf{P}$ depends nonlinearly on a parameter $\mu$ and solutions are sought for many values of this parameter. We show that the compact rational…
We present JAX-FVM, an open-source, fully differentiable finite volume method (FVM) for the two-dimensional compressible Euler and Navier-Stokes equations on unstructured triangular meshes. The solver is written entirely in JAX, so that…
We study iterative linearized finite element methods for the numerical approximation of semilinear elliptic boundary value problems with nonlinear reaction terms of asymptotically linear growth. Our approach reaches considerably beyond the…
We present a framework for generating nonconforming triangular meshes with multiple discretization layers. The framework exploits characteristic structural properties of meshes produced by a frontal Delaunay algorithm with uniform element…
We propose an optimized stencil strategy for the Generalized Finite Difference Method (GFDM) applied to non-linear problems. We take advantage of the flexibility of GFDM to engineer specific stencils by agglomerating nodes and balancing…
Our objective is to solve large systems of ordinary differential equations (ODEs) commonly used to model biological processes. These equations are typically nonlinear, complex, and high-dimensional. In computational biology, such ODEs are…
In this paper, we derive a new model to simulate the incompressible resistive magnetohydrodynamic (MHD) free surface flow. A thermodynamically consistent diffuse interface method is adopted to characterize the moving interface in the…
This paper focuses on the analysis of the tensor equation $\mathcal{A\ltimes X\ltimes B=C}$, formulated via the semi tensor product with t-product. For the unknown vector $\mathcal{X}$, we establish a necessary and sufficient condition that…
The projected linear system solver (PLSS), by incrementally appending columns to a random or deterministic sketching matrix, provides an attractive finite termination property for consistent linear systems. Nevertheless, a critical…
We analyze a novel locking-free mixed formulation for the elasticity eigenvalue problem in both two and three dimensions, expressed exclusively in terms of the pseudostress tensor. An important feature of this formulation is that it does…
In this paper we consider explicit Runge--Kutta (RK) methods for the numerical solution of Initial Value Problems (IVPs) in differential equations in which the last function evaluation in a step is reused, substituting the first evaluation…
We study fixed-tolerance low-rank approximation in the matrix-free setting, where a matrix or linear operator $\mathbf{A}$ is accessible only through matrix-vector products and its rank must be determined adaptively to meet a prescribed…
In recent years, quantum linear system algorithms have been applied to partial differential equations (PDEs), particularly in high-dimensional settings, demonstrating an exponential speedup in dimension. Concurrently, randomized and…
We study the scattering of time-harmonic electromagnetic waves by periodic layered gratings, modelled by the 2D Helmholtz equation. The periodic obstacle may include penetrable and impenetrable regions, and consists of a finite number of…
Low-rank tensor methods are an important tool in the numerical treatment of equations with a high-dimensional state space. Nearest neighbor interaction systems like the Ising model or more general Markov jump processes, as well as 1D…
Global space-time spectral methods give spectral accuracy in time but typically require the whole space-time history to be resolved and stored on a single tensor-product domain $T \times \Omega$. We record that in an endpoint-benign…