A monotone combination scheme of diffusion equations on polygonal meshes

Abstract

We present a novel monotone scheme which is a combination of linear scheme and nonlinear monotone scheme for solving diffusion problems on general polygonal meshes. It will be called as a combination scheme and consists of two steps. Firstly, a second‐order accurate linear scheme is used to obtain an approximate solution. Secondly, a nonlinear monotone scheme is used to solve the diffusion equation, where the unknowns in the nonlinear coefficient of the nonlinear monotone scheme are taken as the approximate solution of the linear scheme above, i.e., a linearized monotone scheme is obtained. So the combination scheme does not require nonlinear iterations for solving linear diffusion problems, moreover, it benefits from the accuracy and efficiency of the linear scheme, as well as the monotonicity of nonlinear monotone scheme. We also analyze some properties satisfied by the combination scheme, such as conservation, stability, monotonicity and convergence. Numerical results are presented to show the performance of the monotone combination scheme on distorted meshes, especially some numerical comparisons among our combination scheme with some existing linear and nonlinear monotone schemes are given.

Arbitrary-order monotonic finite-volume schemes for 1D elliptic problems

“…Proposition 2.3. Assume that u > 0, the matrix A(u) defined by ( 30) and ( 31) through (34), or (35) through (38), or (39) through (42) depending on the boundary conditions, with s i+ 1 2 = 0, is such that A T (u) is a strict Mmatrix.…”

“…Assume that f ≥ 0, g ≥ 0, and either f L 2 (Ω) > 0, g(0) > 0 or g(1) > 0. Let A and b be defined by ( 30) and ( 31) through (34), or (35) through (38), or (39) through (42), depending on the boundary conditions. Assume that we have applied the symmetrization procedure defined in Section 1.4.…”

“…In [4], a non-linear solver based on an iterative resolution of two problems is described, the primary unknowns of one problem being the secondary unknowns of the other one. The works [38,24] explain how to build monotonic schemes without relying on secondary unknowns. In [22,25,30], maximum principle preserving schemes are proposed.…”

Monotonic Diamond and DDFV Type Finite-Volume Schemes for 2D Elliptic Problems

“…In [38] the requirement of positive secondary unknowns is relaxed. The works [31,50] explain how to build monotonic schemes without relying on secondary unknowns. In [29,32,37], maximum principle preserving schemes are proposed.…”

Arbitrary Order Monotonic Finite-Volume Schemes for 2d Elliptic Problems

“…This is why a large amount of work has been devoted to the design of monotone schemes since the seminal works of [5,26]. Among other publications, let us cite recent works [11,12,30,32,34,36,37,40] and references therein about this topic. However, none of these methods is arbitrarily high-order accurate.…”