High-order Complete Flux Schemes for Convection-Diffusion Equations on Arbitrary Subdivisions

Jul 16, 2026

Develops a complete-flux finite volume method for convection–diffusion equations on arbitrary 2D and 3D meshes and for discrete spaces of any polynomial degree. Exact normal numerical fluxes are derived from the underlying PDE, splitting into a homogeneous Scharfetter–Gummel part and an inhomogeneous Green’s-function contribution capturing tangential flux and sources. The formulation is exactly equivalent to the continuous model and provides high-order accuracy without correction or stabilization, achieving optimal second-order behavior for linear spaces and optimal L2 convergence for higher-degree spaces.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages23

Reading Time58 min

Introduction

Background and motivation from convection-dominated problems

Finite volume methods and Scharfetter–Gummel-type exponential fitting

Complete flux methodology and prior extensions

Challenges for high-order finite volume schemes on arbitrary meshes

arXiv :2607 .08422v1 [math .NA] 9 Jul 2026 HIGH-ORDER COMPLETE FLUX SCHEMES FOR CONVECTION-DIFFUSION EQUATIONS ON ARBITRARY SUBDIVISIONS PENG YANG∗, WENYU LEI∗ , AND LIWEI XU∗ Abstract. We develop a complete ﬂux ﬁnite volume method for convection-diﬀusion equations that works on arbitrary meshes in two and three dimensions and for discrete spaces of any polynomial degree. Unlike standard ﬁnite volume discretizations, where the numerical ﬂux is directly approximated from the ﬂux deﬁnition, we derive the exact normal ﬂux across each control volume edge/face from the underlying PDE. This exact ﬂux splits naturally into a homogeneous part (the classical Scharfetter-Gummel ﬂux) and an inhomogeneous part expressed via a Green’s function that incorporates the tangential ﬂux and the source term. The resulting formulation is exactly equivalent to the continuous equation and, once the discrete space is chosen, yields high-order schemes without any correction or stabilization. For piecewise linear spaces, the scheme achieves optimal second-order accuracy in convectiondominated regimes and can preserve positivity on moderately coarse meshes. For quadratic spaces, standard ﬁnite volume methods, based on the Lagrange elements or B-splines, fail to attain optimal L2 convergence unless the control volume mesh is specially designed. The proposed complete ﬂux scheme, however, always achieves optimal L2 convergence independently of the control volume mesh. Numerical experiments in two and three dimensions conﬁrm the robustness and optimal accuracy of the approach. Key words. complete ﬂux, ﬁnite volume, convection-diﬀusion, high-order, B-spline MSC codes. 65N08, 65D07, 65L11 1. Introduction. Steady-state convection-diﬀusion equations arise in numerous physical and engineering applications, including semiconductor device simulation [24], groundwater contaminant transport [30], heat and mass transfer in ﬂuid ﬂows [20], and atmospheric pollutant dispersion [23] . When the convection coeﬃcient dominates over diﬀusion, the problem becomes singularly perturbed and its numerical approximation is notoriously challenging: standard Galerkin methods produce spurious oscillations unless the mesh is excessively reﬁned [3], while simple upwinding introduces excessive numerical diﬀusion [20] . Finite volume methods are attractive for these problems due to their local conservation property and their natural ability to handle discontinuous coeﬃcients and complex geometries [1, 7, 10, 15, 16] . A particularly inﬂuential class of methods for convection-dominated problems originates from the work of Scharfetter and Gummel [22] in semiconductor physics. The Scharfetter–Gummel (SG) scheme solves a local one-dimensional boundary value problem on the interval between two adjacent grid points, assuming constant coeﬃcients and neglecting the source term. The resulting numerical ﬂux, expressed in terms of the Bernoulli function, resolves sharp layers without spurious oscillations even on relatively coarse meshes. Building on the same exponential ﬁtting technique, related methods have been developed in various numerical frameworks, including ﬁnite element and ﬁnite volume discretizations [21], and recently discontinuous Galerkin methods [14] . A more systematic generalization of this idea is the complete ﬂux (CF) scheme proposed by Thiart [27] and further investigated by Ten Thije Boonkkamp and co-workers [19, 25, 26], who derived the CF formulation by retaining the source term contribution in the Scharfetter–Gummel ﬂux. In multiple spatial dimensions, the tangential ﬂux divergence naturally enters ∗ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China ([pyang@uestc.edu.cn](pyang@uestc.edu.cn), [wenyu.lei@uestc.edu.cn](wenyu.lei@uestc.edu.cn), [xul@uestc.edu.cn](xul@uestc.edu.cn)). 1 This manuscript is for review purposes only. 2 P. YANG, W. LEI AND L. XU the local ODE as part o

FAQs

What distinguishes the complete-flux method from standard finite volume discretizations for convection–diffusion equations?

Standard methods approximate the numerical flux directly from a flux definition. The complete-flux approach derives the exact normal flux across each control-volume edge/face from the original PDE, then splits it into a homogeneous Scharfetter–Gummel part and an inhomogeneous Green’s-function part.

How does the proposed method incorporate source terms and tangential effects?

The inhomogeneous flux term is expressed using a Green’s function that incorporates the tangential flux divergence and the source contribution, embedding these effects into the numerical flux construction.

Why are high-order schemes difficult for quadratic and higher-degree polynomial spaces, and how does the proposal address it?

Standard finite volume methods fail to achieve optimal L2 convergence on arbitrary dual/control-volume meshes for quadratic (and higher even-degree) spaces. The proposed complete-flux scheme attains optimal L2 convergence independently of how the control-volume mesh is chosen, supported by numerical experiments in 2D and 3D.

Jul 16, 2026

Upload by Valentina