arXiv subjects

Peter Minev

Publications and source records attributed to Peter Minev.

10 recordsLinked to original sources

arXiv2026

Efficient and simple fourth-order compact finite difference methods for convection-diffusion-reaction equations on arbitrary curved domains

In this paper, we discuss the 2D convection-diffusion-reaction equation with variable smooth coefficients and the Dirichlet boundary condition on a complicated, thin, and curved domain. We propose the fourth-order compact FDM at every grid point with the uniform Cartesian mesh. For the regular stencil center, we utilize the fourth-order compact 9-point FDM to approximate the solution. According to the preliminary analysis, we use vertical and horizontal transformations to derive fourth-order compact FDMs in 10 cases for all irregular stencil centers. To obtain the left-hand side of the stencil of the fourth-order FDM in each case, we only need to solve an at most $6 \times 24$ linear system which is presented with the explicit formula. The right-hand side of the FDM is constructed in explicit expression for any irregular stencil centers too. To achieve the fourth-order consistency, up to second-order partial derivatives of convection, diffusion, reaction, and source terms are used for the FDM at the regular stencil center, and the FDM at an irregular stencil center only requires first-order partial derivatives of convection, diffusion, reaction, and source terms, and up to third-order derivatives of the Dirichlet boundary function and the parametric expression of the boundary curve. We test challenging domains with 100-leaf, high-curvature, high-frequency, sharply varying, and nearly overlapping boundary curves, the proposed FDM produces the high accuracy and the stable fourth-order convergence rate in $l_2$ and $l_{\infty}$ norms. All stencils of our FDMs have a simple desired structure by only keeping grid points inside $\Omega$ in the standard compact 9-point stencil for both regular stencils and boundary stencils, but without assuming any information outside the domain $\Omega$.

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arXiv2023

Sixth-Order Hybrid Finite Difference Methods for Elliptic Interface Problems with Mixed Boundary Conditions

In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem $-\nabla \cdot( a\nabla u)=f$ in $\Omega\backslash \Gamma$, where $\Gamma$ is a smooth interface inside $\Omega$. The variable scalar coefficient $a>0$ and source $f$ are possibly discontinuous across $\Gamma$. The hybrid FDMs utilize a $9$-point compact stencil at any interior regular points of the grid and a $13$-point stencil at irregular points near $\Gamma$. For interior regular points away from $\Gamma$, we obtain a sixth-order $9$-point compact FDM satisfying the sign and sum conditions for ensuring the M-matrix property. We also derive sixth-order compact ($4$-point for corners and $6$-point for edges) FDMs satisfying the sign and sum conditions for the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. Thus, for the elliptic problem without interface (i.e., $\Gamma$ is empty), our compact FDM has the M-matrix property for any mesh size $h>0$ and consequently, satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. For irregular points near $\Gamma$, we propose fifth-order $13$-point FDMs, whose stencil coefficients can be effectively calculated by recursively solving several small linear systems. Theoretically, the proposed high order FDMs use high order (partial) derivatives of the coefficient $a$, the source term $f$, the interface curve $\Gamma$, the two jump functions along $\Gamma$, and the functions on $\partial \Omega$. Numerically, we always use function values to approximate all required high order (partial) derivatives in our hybrid FDMs without losing accuracy. Our numerical experiments confirm the sixth-order convergence in the $l_{\infty}$ norm of the proposed hybrid FDMs for the elliptic interface problem.

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arXiv2022

Compact 9-Point Finite Difference Methods with High Accuracy Order and/or M-Matrix Property for Elliptic Cross-Interface Problems

In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In the present paper we propose a fourth-order 9-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. Furthermore, we show that the resulting linear system for the special case has an M-matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical experiments demonstrate the fourth (for the general case) and sixth (for the special case) accuracy orders of the proposed schemes. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an M-matrix. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem.

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arXiv2022

Hybrid Finite Difference Schemes for Elliptic Interface Problems with Discontinuous and High-Contrast Variable Coefficients

For elliptic interface problems with discontinuous coefficients, the maximum accuracy order for compact 9-point finite difference scheme in irregular points is three [7]. The discontinuous coefficients usually have abrupt jumps across the interface curve in the porous medium of realistic problems, causing the pollution effect of numerical methods. So, to obtain a reasonable numerical solution of the above problem, the higher order scheme and its effective implementation are necessary. In this paper, we propose an efficient and flexible way to achieve the implementation of a hybrid (9-point scheme with sixth order accuracy for interior regular points and 13-point scheme with fifth order accuracy for interior irregular points) finite difference scheme in uniform meshes for the elliptic interface problems with discontinuous and high-contrast piecewise smooth coefficients in a rectangle $\Omega$. We also derive the $6$-point and $4$-point finite difference schemes in uniform meshes with sixth order accuracy for the side points and corner points of various mixed boundary conditions (Dirichlet, Neumann and Robin) of elliptic equations in a rectangle. Our numerical experiments confirm the flexibility and the sixth order accuracy in $l_2$ and $l_{\infty}$ norms of the proposed hybrid scheme.

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arXiv2021

A High Order Compact Finite Difference Scheme for Elliptic Interface Problems with Discontinuous and High-Contrast Coefficients

The elliptic interface problems with discontinuous and high-contrast coefficients appear in many applications and often lead to huge condition numbers of the corresponding linear systems. Thus, it is highly desired to construct high order schemes to solve the elliptic interface problems with discontinuous and high-contrast coefficients. Let $\Gamma$ be a smooth curve inside a rectangular region $\Omega$. In this paper, we consider the elliptic interface problem $-\nabla\cdot (a \nabla u)=f$ in $\Omega\setminus \Gamma$ with Dirichlet boundary conditions, where the coefficient $a$ and the source term $f$ are smooth in $\Omega\setminus \Gamma$ and the two nonzero jump condition functions $[u]$ and $[a\nabla u\cdot \vec{n}]$ across $\Gamma$ are smooth along $\Gamma$. To solve such elliptic interface problems, we propose a high order compact finite difference scheme for numerically computing both the solution $u$ and the gradient $\nabla u$ on uniform Cartesian grids without changing coordinates into local coordinates. Our numerical experiments confirm the fourth order accuracy for computing the solution $u$, the gradient $\nabla u$ and the velocity $a \nabla u$ of the proposed compact finite difference scheme on uniform meshes for the elliptic interface problems with discontinuous and high-contrast coefficients.

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arXiv2021

Sixth Order Compact Finite Difference Scheme for Poisson Interface Problem with Singular Sources

Let $\Gamma$ be a smooth curve inside a two-dimensional rectangular region $\Omega$. In this paper, we consider the Poisson interface problem $-\nabla^2 u=f$ in $\Omega\setminus \Gamma$ with Dirichlet boundary condition such that $f$ is smooth in $\Omega\setminus \Gamma$ and the jump functions $[u]$ and $[\nabla u\cdot \vec{n}]$ across $\Gamma$ are smooth along $\Gamma$. This Poisson interface problem includes the weak solution of $-\nabla^2 u=f+g\delta_\Gamma$ in $\Omega$ as a special case. Because the source term $f$ is possibly discontinuous across the interface curve $\Gamma$ and contains a delta function singularity along the curve $\Gamma$, both the solution $u$ of the Poisson interface problem and its flux $\nabla u\cdot \vec{n}$ are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils. The coefficient matrix $A$ in the resulting linear system $Ax=b$, following from the proposed scheme, is independent of any source term $f$, jump condition $g\delta_\Gamma$, interface curve $\Gamma$ and Dirichlet boundary conditions. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.

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arXiv2020

An efficient algorithm for weakly compressible flows in spherical geometries

This study proposes an algorithm for modeling compressible flows in spherical shells in nearly incompressible and weakly compressible regimes based on an implicit direction splitting approach. The method retains theoretically expected convergence rates and remains stable for extremely small values of the characteristic Mach number. The staggered spatial discretization on the MAC stencil, commonly used in numerical methods for incompressible Navier-Stokes equations, was found to be convenient for the discretization of the compressible Navier-Stokes equations written in the non-conservative form in terms of the primitive variables. This approach helped to avoid the high-frequency oscillations without any artificial stabilization terms. Nonlinear Picard iterations with the splitting error reduction were also implemented to allow one to obtain a solution of the fully nonlinear system of equations. These results, alongside excellent parallel performance, prove the viability of the direction splitting approach in large-scale high-resolution high-performance simulations of atmospheric and oceanic flows.

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arXiv2019

A direction splitting scheme for Navier-Stokes-Boussinesq system in spherical shell geometries

This paper introduces a formally second-order direction-splitting method for solving the incompressible Navier-Stokes-Boussinesq system in a spherical shell region. The equations are solved on overset Yin-Yang grids, combined with spherical coordinate transforms. This approach allows to avoid the singularities at the poles and keeps the grid size relatively uniform. The downside is that the spherical shell is subdivided into two equally sized, overlapping subdomains that requires the use of Schwarz-type iterations. The temporal second order accuracy is achieved via an Artificial Compressibility (AC) scheme with bootstrapping. The spatial discretization is based on second order finite differences on the Marker-And-Cell (MAC) stencil. The entire scheme is implemented in parallel using a domain decomposition iteration, and a direction splitting approach for the local solves. The stability, accuracy and weak scalability of the method is verified on a manufactured solution of the Navier-Stokes-Boussinesq system and on the Landau solution of the Navier-Stokes equations on the sphere.

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arXiv2016

Splitting schemes for unsteady problems involving the grad-div operator

In this paper we consider various splitting schemes for unsteady problems containing the grad-div operator. The fully implicit discretization of such problems would yield at each time step a linear problem that couples all components of the solution vector. In this paper we discuss various possibilities to decouple the equations for the different components that result in unconditionally stable schemes. If the spatial discretization uses Cartesian grids, the resulting schemes are Locally One Dimensional (LOD). The stability analysis of these schemes is based on the general stability theory of additive operator-difference schemes developed by Samarskii and his collaborators. The results of the theoretical analysis are illustrated on a 2D numerical example with a smooth manufactured solution.

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arXiv2016

High-order time stepping for the Navier-Stokes equations with minimal computational complexity

In this paper we present extensions of the schemes proposed in \cite{GM14} that lead to a decoupling of the velocity components in the momentum equation. The new schemes reduce the solution of the incompressible Navier-Stokes equations to a set of classical uncoupled parabolic problems for each Cartesian component of the velocity. The pressure is explicitly recovered after the velocity is computed.

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