The field of numerical methods for partial differential equations is rapidly advancing, with a focus on developing innovative and efficient methods for solving complex problems. Recent developments have centered around improving the accuracy and stability of existing methods, as well as exploring new approaches such as deep neural networks and low-rank solvers. Notably, researchers have made significant progress in designing methods that preserve energy and other physical quantities, which is crucial for simulating real-world phenomena. Furthermore, there is a growing interest in developing methods that can handle stochastic and uncertain systems, which is essential for modeling complex systems in various fields. Some noteworthy papers in this area include the development of a low-rank solver for the Stokes-Darcy model with random hydraulic conductivity and the proposal of a novel generalized low-rank approximation of large-scale stiffness matrices. Additionally, researchers have made significant progress in establishing inverse inequalities for kernel-based approximation spaces, which has important implications for a wide range of applications.
Advances in Numerical Methods for Partial Differential Equations
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Solitary-wave solutions of the fractional nonlinear Schr\"{o}dinger equation. II. A numerical study of the dynamics
Full Vectorial Maxwell Equations with Continuous Angular Indices
Automated $h$-adaptivity for finite element approximations of the Falkner-Skan equation
Diffusive behavior of transport noise on $\mathbb{S}^2$
Spline Shallow Water Moment Equations
Mixed Finite Element Method for a Hemivariational Inequality of Stationary convective Brinkman-Forchheimer Extended Darcy equations
H(curl)-based approximation of the Stokes problem with weakly enforced no-slip boundary conditions
Filtering and 1/3 Power Law for Optimal Time Discretisation in Numerical Integration of Stochastic Differential Equations
Two operator splitting methods for three-dimensional stochastic Maxwell equations with multiplicative noise
Error Estimates of Semi-Lagrangian Schemes for Diffusive Conservation Laws
Unconditional energy dissipation of Strang splitting for the matrix-valued Allen-Cahn equation
Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions
Deep Neural Networks with General Activations: Super-Convergence in Sobolev Norms
An asymptotic-preserving active flux scheme for the hyperbolic heat equation in the diffusive scaling
A low-rank solver for the Stokes-Darcy model with random hydraulic conductivity and Beavers-Joseph condition
The domain-of-dependence stabilization for cut-cell meshes is fully discretely stable
Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds
Numerical analysis of the stochastic Navier-Stokes equations