The field of numerical methods for partial differential equations is rapidly evolving, with a focus on developing innovative and efficient algorithms for solving complex problems. Recent research has emphasized the importance of preserving physical properties, such as stationarity and energy conservation, in numerical discretizations. High-order methods, such as finite element and finite difference methods, have been developed to improve accuracy and efficiency. Additionally, new formulations and techniques, such as virtual element methods and integral equation methods, have been proposed to tackle challenging problems. Notable papers in this area include the development of stationarity preserving nodal Finite Element methods for multi-dimensional linear hyperbolic balance laws and the creation of a high-order, compact, and symmetric Finite Difference Method for the variable Poisson equation on a d-dimensional hypercube. The development of a mass-lumped Virtual Element Method with strong stability-preserving Runge-Kutta time stepping for two-dimensional parabolic problems has also shown promising results. Overall, these advances have the potential to significantly impact the field of numerical analysis and its applications in various areas of science and engineering.
Advances in Numerical Methods for Partial Differential Equations
Sources
Stationarity preserving nodal Finite Element methods for multi-dimensional linear hyperbolic balance laws via a Global Flux quadrature formulation
General Order Virtual Element Approximation for the Smagorinsky turbulence model
Fourier-Galerkin method for scattering poles of sound soft obstacles
A Variational Method for Conformable Fractional Equations Using Rank-One Updates
High order well-balanced and total-energy-conserving local discontinuous Galerkin methods for compressible self-gravitating Euler equations
High-order, Compact, and Symmetric Finite Difference Methods for a $d$-Dimensional Hypercube
A note on spectral Monte-Carlo method for fractional Poisson equation on high-dimensional ball
Convergence of the Immersed Boundary Method for an Elastically Bound Particle Immersed in a 2D Navier-Stokes Fluid Fluid
Algorithm for constructing optimal explicit finite-difference formulas in the Hilbert space
Mass-Lumped Virtual Element Method with Strong Stability-Preserving Runge-Kutta Time Stepping for Two-Dimensional Parabolic Problems
An Integral Equation Method for Linear Two-Point Boundary Value Systems
A Higher-Order Time Domain Boundary Element Formulation based on Isogeometric Analysis and the Convolution Quadrature Method
Geometric Queries on Closed Implicit Surfaces for Walk on Stars