arXiv subjects

Qiwei Feng

Publications and source records attributed to Qiwei Feng.

14 recordsLinked to original sources

arXiv2026

Analysis of Compact Symmetric Finite Difference Methods with Highest Possible Orders for Elliptic Equations

For the $d$-dimensional variable-coefficient Poisson problem $-\nabla \cdot(a\nabla u)=f$ subject to Dirichlet boundary conditions, we seek compact (i.e., $3^d$-point) symmetric finite difference methods (FDMs) of the highest possible consistency orders, which yield symmetric positive definite (SPD) linear systems so that many fast solvers (e.g., the conjugate gradient method) can be used. Although 2nd-order compact symmetric FDMs for variable-coefficient Poisson equations are known, the construction of higher-order counterparts has, to the best of our knowledge, remained an open problem. This is largely because all stencils must satisfy highly specific interdependent relations and high-order consistency imposes numerous local technical conditions. This paper provides a comprehensive characterization of the highest consistency order that a $d$-dimensional compact symmetric FDM can achieve for variable-coefficient Poisson equations on a uniform grid. For the 1D case, we prove that our symmetric compact FDMs on a uniform grid can achieve arbitrary consistency orders, yield tridiagonal SPD matrices, and converge at the corresponding orders in the $\infty$-norm and the weighted $2$-norm. For any $d$-dimensional case with $d\ge 2$, we prove that compact (not necessarily symmetric) FDMs can achieve 6th-order consistency and that no higher consistency order is possible. If $\tfrac{|\nabla a|^2}{a^2}-\tfrac{2Δa}{a}$ is nonconstant, $d$-dimensional compact symmetric FDMs can achieve 4th-order consistency, but no higher. On the other hand, if $\tfrac{|\nabla a|^2}{a^2}-\tfrac{2Δa}{a}$ is constant, then $d$-dimensional compact symmetric FDMs can achieve 6th-order consistency, which is the highest possible order. For $d=2,3,4$ and $M=4,6$, we prove that our proposed FDMs produce SPD matrices, and achieve $M$th-order convergence in the weighted 2-norm and the $\infty$-norm.

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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 $Ω$ in the standard compact 9-point stencil for both regular stencils and boundary stencils, but without assuming any information outside the domain $Ω$.

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arXiv2026

Fourth-order compact finite difference methods for 2D and 3D nonlinear convection-diffusion-reaction equations

In this paper, we first consider linear 2D and 3D convection-diffusion-reaction equations $-\nabla\cdot (κ\nabla u) + {\bm v} \cdot \nabla u + λu = ϕ$ and $u_t - \nabla\cdot (κ\nabla u) + {\bm v} \cdot \nabla u + λu = ϕ$, where all $κ>0, {\bm v}, λ, ϕ$ are smooth variable functions. We derive fourth-order compact 9-point (2D) and 19-point (3D) finite difference methods (FDMs) to solve linear time-independent equations. As derivations of high-order compact FDMs are very complicated and involve cumbersome notation (especially in 3D), it is usually difficult for readers not specializing in high-order FDMs to follow derivations and replicate numerical results. In this paper, we observe interesting and novel expressions of stencils of high-order FDMs which introduce new restrictions of stencils to help construct compact fourth-order FDMs (2D and 3D) with easy, explicit, and short expressions. These simple stencils make the analysis of the truncation error easy for readers to understand and facilitate implementing proposed FDMs directly. For linear unsteady equations, we apply Crank-Nicolson (CN), BDF3, BDF4 methods with above compact FDMs to compute numerical solutions. Finally, we discuss nonlinear convection-diffusion-reaction equations in 2D and 3D, i.e., each function of $κ(u)>0, {\bm v}(u), λ(u)$ depends on the solution $u$. We linearize nonlinear equations by the fixed point method (iterative method) and use above simple fourth-order compact FDMs to solve linearized equations (unsteady equations also utilize CN, BDF3, and BDF4 methods). Each of proposed FDMs in 2D and 3D for linear, nonlinear, steady, and unsteady equations satisfies the discrete maximum principle and forms an M-matrix for the sufficiently small $h$, if the function $λ$ is nonnegative.

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arXiv2025

Fourth-Order Compact FDMs for Steady and Time-Dependent Nonlinear Convection-Diffusion Equations

In this paper, we discuss the steady and time-dependent nonlinear convection-diffusion (advection-diffusion) equations with the Dirichlet boundary condition. For the steady nonlinear equation, we use an iteration method to reformulate the nonlinear equation into its linear counterpart, and derive a fourth-order compact 9-point finite difference method (FDM) to solve the reformulated equation on a uniform Cartesian grid. To increase the accuracy, we modify the FDM to reduce the pollution effect. The linear system of the FDM generates an M-matrix, provided the mesh size $h$ is sufficiently small. For the time dependent nonlinear equation, we discrete the temporal domain using the Crank-Nicolson (CN), BDF3, BDF4 time stepping methods, and apply a similar iterative method to rewrite the nonlinear equation as the same linear convection-diffusion equation. Then we propose the second-order to fourth-order compact 9-point FDMs with the reduced pollution effects on a uniform Cartesian grid. We prove that all FDMs satisfy the discrete maximum principle for sufficiently small $h$. Several examples with the variable and time-dependent diffusion coefficients and challenging nonlinear terms (not limited to the Burgers equation) are provided to verify the accuracy and the desired convergence rates in the $l_2$ and $l_{\infty}$ norms in space and time. We also compare our second-order CN method with the third-order BDF3 method and the discontinuous Galerkin (DG) method, and the numerical results demonstrate that our FDM with the coarse time step generates the small error. Especially, if the same BDF3 scheme is applied, our error is 1.6\% of that obtained from the DG method. The proposed methods can be easily extended to a 3D spatial domain and more general nonlinear convection-diffusion-reaction equations.

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arXiv2025

A High-Order, Pressure-Robust, and Decoupled Finite Difference Method for the Stokes Problem

In this paper, we consider the Stokes problem with Dirichlet boundary conditions and the constant kinematic viscosity $ν$ in an axis-aligned domain $Ω$. We decouple the velocity $\bm u$ and pressure $p$ by deriving a novel biharmonic equation in $Ω$ and third-order boundary conditions on $\partialΩ$. In contrast to the fourth-order streamfunction approach, our formulation does not require $Ω$ to be simply connected. For smooth velocity fields $\bm u$ in two dimensions, we explicitly construct a finite difference method (FDM) with sixth-order consistency to approximate $\bm u$ at all relevant grid points: interior points, boundary side points, and boundary corner points. The resulting scheme yields two linear systems $A_1u^{(1)}_h=b_1$ and $A_2u^{(2)}_h=b_2$, where $A_1,A_2$ are constant matrices, and $b_1,b_2$ are independent of the pressure $p$ and the kinematic viscosity $ν$. Thus, the proposed method is pressure- and viscosity-robust. To accommodate velocity fields with less regularity, we modify the FDM by removing singular terms in the right-hand side vectors. Once the discrete velocity is computed, we apply a sixth-order finite difference operator to approximate the pressure gradient locally, without solving any additional linear systems. In our numerical experiments, we test both smooth and non-smooth solutions $(\bm u,p)$ in a square domain, a triply connected domain, and an $L$-shaped domain in two dimensions. The results confirm sixth-order convergence of the velocity and pressure gradient in the $\ell_\infty$-norm for smooth solutions. For non-smooth velocity fields, our method achieves the expected lower-order convergence. Moreover, the observed velocity error $\|{\bm u}_h-\bm u\|_{\infty}$ is independent of the pressure $p$ and viscosity $ν$.

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arXiv2025

The convergence proof of the sixth-order compact 9-point FDM for the 2D transport problem

It is widely acknowledged that the convergence proof of the error in the $l_{\infty}$ norm of the high-order finite difference method (FDM) and finite element method (FEM) in 2D is challenging. In this paper, we derive the sixth-order compact 9-point FDM with the explicit stencil for the 2D transport problem with the constant coefficient and the Dirichlet boundary condition in a unit square. The proposed sixth-order FDM forms an M-matrix for the any mesh size $h$ employing the uniform Cartesian mesh. The explicit formula of our FDM also enables us to construct the comparison function with the explicit expression to rigorously prove the sixth-order convergence rate of the maximum pointwise error by the discrete maximum principle. Most importantly, we demonstrate that the sixth-order convergence proof is valid for any mesh size $h$. The numerical results are consistent with sixth-order accuracy in the $l_{\infty}$ norm. Our theoretical convergence proof is clear and the proposed sixth-order FDM is straightforward to be implemented, facilitating the reproduction of our numerical results.

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arXiv2024

Distinct Numerical Solutions for Elliptic Cross-Interface Problems Using Finite Element and Finite Difference Methods

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 coefficients, two homogeneous jump conditions, continuous source terms, and Dirichlet boundary conditions. For brevity, we consider a 2D simplified version where the intersection points of the interface lines coincide with grid points in uniform Cartesian grids. Our findings reveal interesting and important results: (1) When the coefficient functions exhibit either high jumps with low-frequency oscillations or low jumps with high-frequency oscillations, the finite element method and finite difference method yield similar numerical solutions. (2) However, when the interface problems involve high-contrast and high-frequency coefficient functions, the numerical solutions obtained from the finite element and finite difference methods differ significantly. Given that the widely studied SPE10 benchmark problem (see https://www.spe.org/web/csp/datasets/set02.htm) typically involves high-contrast and high-frequency permeability due to varying geological layers in porous media, this phenomenon warrants attention. Furthermore, this observation is particularly important for developing multiscale methods, as reference solutions for these methods are usually obtained using the standard second-order finite element method with a fine mesh, and analytical solutions are not available. We provide sufficient details to enable replication of our numerical results, and the implementation is straightforward. This simplicity ensures that readers can easily confirm the validity of our findings.

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arXiv2024

A Derivative-Orthogonal Wavelet Multiscale Method for 1D Elliptic Equations with Rough Diffusion Coefficients

In this paper, we investigate 1D elliptic equations $-\nabla\cdot (a\nabla u)=f$ with rough diffusion coefficients $a$ that satisfy $0<a_{\min}\le a\le a_{\max}<\infty$ and $f\in L_2(Ω)$. To achieve an accurate and robust numerical solution on a coarse mesh of size $H$, we introduce a derivative-orthogonal wavelet-based framework. This approach incorporates both regular and specialized basis functions constructed through a novel technique, defining a basis function space that enables effective approximation. We develop a derivative-orthogonal wavelet multiscale method tailored for this framework, proving that the condition number $κ$ of the stiffness matrix satisfies $κ\le a_{\max}/a_{\min}$, independent of $H$. For the error analysis, we establish that the energy and $L_2$-norm errors of our method converge at first-order and second-order rates, respectively, for any coarse mesh $H$. Specifically, the energy and $L_2$-norm errors are bounded by $2 a_{\min}^{-1/2} \|f\|_{L_2(Ω)} H$ and $4 a_{\min}^{-1}\|f\|_{L_2(Ω)} H^2$. Moreover, the numerical approximated solution also possesses the interpolation property at all grid points. We present a range of challenging test cases with continuous, discontinuous, high-frequency, and high-contrast coefficients $a$ to evaluate errors in $u, u'$ and $a u'$ in both $l_2$ and $l_\infty$ norms. We also provide a numerical example that both coefficient $a$ and source term $f$ contain discontinuous, high-frequency and high-contrast oscillations. Additionally, we compare our method with the standard second-order finite element method to assess error behaviors and condition numbers when the mesh is not fine enough to resolve coefficient oscillations. Numerical results confirm the bounded condition numbers and convergence rates, affirming the effectiveness of our approach.

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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 $Ω\backslash Γ$, where $Γ$ is a smooth interface inside $Ω$. The variable scalar coefficient $a>0$ and source $f$ are possibly discontinuous across $Γ$. 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 $Γ$. For interior regular points away from $Γ$, 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., $Γ$ 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 $Γ$, 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 $Γ$, the two jump functions along $Γ$, and the functions on $\partial Ω$. 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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arXiv2023

Sixth Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect

Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the pollution effect). High order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a high order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. Our method achieves a sixth order consistency for a constant wavenumber, and a fifth order consistency for a piecewise constant wavenumber. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several state-of-the-art finite difference schemes, particularly in the critical pre-asymptotic region where $\textsf{k} h$ is near $1$ with $\textsf{k}$ being the wavenumber and $h$ the mesh size.

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arXiv2023

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 $Ω$. 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 $Γ$ be a smooth curve inside a rectangular region $Ω$. In this paper, we consider the elliptic interface problem $-\nabla\cdot (a \nabla u)=f$ in $Ω\setminus Γ$ with Dirichlet boundary conditions, where the coefficient $a$ and the source term $f$ are smooth in $Ω\setminus Γ$ and the two nonzero jump condition functions $[u]$ and $[a\nabla u\cdot \vec{n}]$ across $Γ$ are smooth along $Γ$. 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 $Γ$ 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 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 $gδ_Γ$, interface curve $Γ$ 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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