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 similar to the Ghost Fluid Method (GFM). The GFM relies on corrections applied on nodes located across the interface for discretization stencils that straddle the interface. If the corrections are solution-independent, they can be moved to the right-hand-side (RHS) of the equations, producing a problem with the same linear system as if there were no jumps, only with a different RHS. However, achieving high accuracy is very hard (if not impossible) with the "standard" approaches used to compute the GFM correction terms.
In this paper we generalize the GFM correction terms to a correction function, defined on a band around the interface. This function is then shown to be characterized as the solution to a PDE, with appropriate boundary conditions. This PDE can, in principle, be solved to any desired order of accuracy. As an example, we apply this new method to devise a 4th order accurate scheme for the constant coefficients Poisson equation with discontinuities in 2D. This scheme is based on (i) the standard 9-point stencil discretization of the Poisson equation, (ii) a representation of the correction function in terms of bicubics, and (iii) a solution of the correction function PDE by a least squares minimization. Several applications of the method are presented to illustrate its robustness dealing with a variety of interface geometries, its capability to capture sharp discontinuities, and its high convergence rate.
In this paper, we study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.
This article aims to reveal the mean-square convergence rate of the backward Euler method (BEM) for a generalized Ait-Sahaliz interest rate model with Poisson jumps. The main difficulty in the analysis is caused by the non-globally Lipschitz drift and diffusion coefficients of the model. We show that the BEM preserves positivity of the original problem. Furthermore, we successfully recover the mean-square convergence rate of order one-half for the BEM. The theoretical findings are accompanied by several numerical examples.
In this paper, we present a second-order accurate finite-difference method for solving convectiondiffusion equations with interfacial jumps on a moving interface. The proposed method is constructed under a semi-Lagrangian framework for convection-diffusion equations; a novel interpolation scheme is developed in the presence of jump conditions. Combined with a second-order ghost fluid method [3], a sharp capturing method with a first-order local truncation error near the interface and second-order truncation error away from the interface is developed for the convectiondiffusion equation. In add…
Almost all materials are anisotropic. In this paper, interface relations of anisotropic elliptic partial differential equations involving discontinuities across interfaces are derived in two and three dimensions. Compared with isotropic cases, the invariance of partial differential equations and the jump conditions under orthogonal coordinates transformation is not valid anymore. A systematic approach to derive the interface relations is established in this paper for anisotropic elliptic interface problems, which can be important for deriving high order accurate numerical methods.
The truncated EM method for stochastic differential equations with Poisson jumps
Nanjing University of Science and Technology（南京理工大学）；Anhui Polytechnic University（安徽工业大学）；Shanghai Normal University（上海师范大学）；University of Strathclyde（思克莱德大学）
Generalized Ait-Sahalia-type interest rate model with Poisson jumps and convergence of the numerical approximation
School of Science, Nanjing University of Science and Technology（南京理工大学理学院）；School of Mathematics and Physics, Anhui Polytechnic University（安徽工程大学数理学院）；Glorious Sun School of Business and Management, Donghua University（东华大学旭日工商管理学院）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）
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 solve Poisson's equation in the presence of discontinuities to high order of accuracy, while using a compact discretization stencil. A key concept behind the CFM is enforcing the jump conditions in a least squares sense. This concept requires computing integrals over sections of the interface, which is a challenge in 3-D when only an implicit representation of t…
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 enrichment IFE piecewise polynomial functions constructed by solving local Cauchy problems on interface elements. The proposed IFE method is based on a discontinuous Galerkin formulation on interface elements and a continuous Galerkin formulation on non-interface elements. This $p^{th}$-degree IFE method is proved to converge optimally under mesh refinement. In a…
We introduce a mathematically rigorous formulation for a nonlocal interface problem with jumps and propose an asymptotically compatible finite element discretization for the weak form of the interface problem. After proving the well-posedness of the weak form, we demonstrate that solutions to the nonlocal interface problem converge to the corresponding local counterpart when the nonlocal data are appropriately prescribed. Several numerical tests in one and two dimensions show the applicability of our technique, its numerical convergence to exact nonlocal solutions, its convergence to the local…
An immersed interface-lattice Boltzmann method (II-LBM) is developed for modelling fluid-structure systems. The key element of this approach is the determination of the jump conditions that are satisfied by the distribution functions within the framework of the lattice Boltzmann method when forces are imposed along a surface immersed in an incompressible fluid. In this initial II-LBM, the discontinuity related to the normal portion of the interfacial force is sharply resolved by imposing the relevant jump conditions using an approach that is analogous to imposing the corresponding pressure jum…
Stochastic evolution equations with compensated Poisson noise are considered in the variational approach with monotone and coercive coefficients. Here the Poisson noise is assumed to be time-homogeneous with $σ$-finite intensity measure on a metric space. By using finite element methods and Galerkin approximations, some explicit and implicit discretizations for this equation are presented and their convergence is proved. Polynomial growth condition and linear growth condition are assumed on the drift operator, respectively for the implicit and explicit schemes.
Let $Γ$ be a smooth curve inside a two-dimensional rectangular region $Ω$. In this paper, we consider the Poisson interface problem $-\nabla^2 u=f$ in $Ω\setminus Γ$ with Dirichlet boundary condition such that $f$ is smooth in $Ω\setminus Γ$ and the jump functions $[u]$ and $[\nabla u\cdot \vec{n}]$ across $Γ$ are smooth along $Γ$. This Poisson interface problem includes the weak solution of $-\nabla^2 u=f+gδ_Γ$ in $Ω$ as a special case. Because the source term $f$ is possibly discontinuous across the interface curve $Γ$ and contains a delta function singularity along the curve $Γ$, both the s…
Singular charge sources in terms of Dirac delta functions present a well-known numerical challenge for solving Poisson's equation. For a sharp interface between inhomogeneous media, singular charges could be analytically treated by fundamental solutions or regularization methods. However, no analytical treatment is known in the literature in case of a diffuse interface of complex shape. This letter reports the first such regularization method that represents the Coulomb potential component analytically by Green's functions to account for singular charges. The other component, i.e., the reactio…
In this paper, we consider a semiconducting device with an active zone made of a single-layer material. The associated Poisson equation for the electrostatic potential (to be solved in order to perform self-consistent computations) is characterized by a surface particle density and an out-of-plane dielectric permittivity in the region surrounding the single-layer. To avoid mesh refinements in such a region, we propose an interface problem based on the natural domain decomposition suggested by the physical device. Two different interface continuity conditions are discussed. Then, we write the c…
Accurate calculation of electrostatic potential and gradient on the molecular surface is highly desirable for the continuum and hybrid modeling of large scale deformation of biomolecules in solvent. In this article a new numerical method is proposed to calculate these quantities on the dielectric interface from the numerical solutions of the Poisson-Boltzmann equation. Our method reconstructs a potential field locally in the least square sense on the polynomial basis enriched with Green's functions, the latter characterize the Coulomb potential induced by charges near the position of reconstru…
We provide unified implementations of the finite element methods involving jumps and averages in Maltab by combing the use of the software package varFEM, including the adaptive finite element methods for the Poisson equation and the $C^0$ interior penalty methods for the biharmonic equation. The design ideas can be extended to other Galerkin-based methods, for example, the discontinuous Galerkin methods and the virtual element methods.
In the present work, we propose a novel hybrid explicit jump immersed interface approach in conjunction with a higher order compact (HOC) scheme for simulating transient complex flows governed by the streamfunction-vorticity ($ψ$-$ζ$) formulation of the Navier-Stokes (N-S) equations for incompressible viscous flows. A new strategy has been adopted for the jump conditions at the irregular points across the interface using Lagrangian interpolation on a Cartesian grid. This approach, which starts with the discretization of parabolic equations with discontinuities in the solutions, source terms an…
Walk-on-Interfaces: A Monte Carlo Estimator for an Elliptic Interface Problem with Nonhomogeneous Flux Jump Conditions and a Neumann Boundary Condition
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 multiple interfaces or with triple-junction points to accommodate interface jump conditions. For non-homogeneous flux jump, we enrich the local approximating spaces by adding up to three local flux basis functions. Numerical experiments are carried out to show that both the Lagrange interpolations and the partial penalized IFEM solutions converge optimally …
