Related papers: A Correction Function Method for the Wave Equation…
In this paper we present a method to treat interface jump conditions for constant coefficients Poisson problems that allows the use of standard "black box" solvers, without compromising accuracy. The basic idea of the new approach is…
We introduce a technique that simplifies the problem of imposing jump conditions on interfaces that are not aligned with a computational grid in the context of the Correction Function Method (CFM). The CFM offers a general framework to…
We propose a high-order FDTD scheme based on the correction function method (CFM) to treat interfaces with complex geometry without increasing the complexity of the numerical approach for constant coefficients. Correction functions are…
We propose high-order FDTD schemes based on the Correction Function Method (CFM) for Maxwell's interface problems with discontinuous coefficients and complex interfaces. The key idea of the CFM is to model the correction function near an…
We present a fast and accurate algorithm to solve Poisson problems in complex geometries, using regular Cartesian grids. We consider a variety of configurations, including Poisson problems with interfaces across which the solution is…
A new finite element method (FEM) using meshes that do not necessarily align with the interface is developed for two- and three-dimensional anisotropic elliptic interface problems with nonhomogeneous jump conditions. The degrees of freedom…
In order to prevent velocity, pressure, and temperature spikes at material discontinuities occurring when the interface-capturing schemes inconsistently simulate compressible multi-material flows(when the specific heats ratio is…
A numerical explicit method to evaluates transient solutions of linear partial differential inhomogeneous equation with constant coefficients is proposed. A general form of the scheme for a specific linear inhomogeneous equation is shown.…
We propose the Compact Coupling Interface Method (CCIM), a finite difference method capable of obtaining second-order accurate approximations of not only solution values but their gradients, for elliptic complex interface problems with…
In this paper, we discuss the second-order finite element method (FEM) and finite difference method (FDM) for numerically solving elliptic cross-interface problems characterized by vertical and horizontal straight lines, piecewise constant…
Conditional flow matching (CFM) stands out as an efficient, simulation-free approach for training flow-based generative models, achieving remarkable performance for data generation. However, CFM is insufficient to ensure accuracy in…
Finite difference schemes for the simulation of elastic waves in materi- als with jump discontinuities are presented. The key feature is the highly accurate treatment of interfaces where media discontinuities arise. The schemes are…
In this article, we introduce a new partially penalized immersed finite element method (IFEM) for solving elliptic interface problems with multi-domains and triple-junction points. We construct new IFE functions on elements intersected with…
In this work, we propose staggered FDTD schemes based on the correction function method (CFM) to discretize Maxwell's equations with embedded perfect electric conductor (PEC) boundary conditions. The CFM uses a minimization procedure to…
This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods…
In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods…
One of the oldest and most studied subject in scientific computing is algorithms for solving partial differential equations (PDEs). A long list of numerical methods have been proposed and successfully used for various applications. In…
In this paper, we propose a novel high order unfitted finite element method on Cartesian meshes for solving the acoustic wave equation with discontinuous coefficients having complex interface geometry. The unfitted finite element method…
This article presents and analyzes a $p^{th}$-degree immersed finite element (IFE) method for elliptic interface problems with nonhomogeneous jump conditions. In this method, jump conditions are approximated optimally by basic IFE and…
The random feature method (RFM) has demonstrated great potential in bridging traditional numerical methods and machine learning techniques for solving partial differential equations (PDEs). It retains the advantages of mesh-free approaches…