Convergence of a positive nonlinear DDFV scheme for degenerate parabolic equations

Abstract

In this work, we carry out the convergence analysis of a positive DDFV method for approximating solutions of degenerate parabolic equations. The basic idea rests upon different approximations of the fluxes on the same interface of the control volume. Precisely, the approximated flux is split into two terms corresponding to the primal and dual normal components. Then the first term is discretized using a centered scheme whereas the second one is approximated in a non evident way by an upstream scheme. The novelty of our approach is twofold: on the one hand we prove that the resulting scheme preserves the positivity and on the other hand we establish energy estimates. Some numerical tests are presented and they show that the scheme in question turns out to be robust and efficient. The accuracy is almost of second order on general meshes when the solution is smooth.

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Acknowledgements

The authors would like to thank the anonymous reviewer for his valuable comment and some precise suggestions that improved the presentation and the quality of this work.

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Correspondence to Marianne Bessemoulin-Chatard.

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This work is supported by: Ministère de l’Enseignement Supérieur, de la Recherche Scientifique et de la Formation des Cadres du Maroc, CNRST and l’Institut Français au Maroc, and Project MoHyCon—ANR-17-CE40-0027-01.

Appendix: Technical lemmas

Appendix: Technical lemmas

Let \({\mathcal {D}}\) be a fixed diamond cell. We define the following \(2\times 2\) matrices

$$\begin{aligned} \mathbb {A}^{{\mathcal {D}}}&= \dfrac{1}{4 m_{{\mathcal {D}}}} \begin{bmatrix} m_{ \sigma }^2 \ &{} m_{ \sigma }m_{\sigma ^*}\\ m_{ \sigma }m_{\sigma ^*}&{} m_{\sigma ^*}^2 \end{bmatrix} {=}{:} \begin{bmatrix} \mathbb {A}^{{\mathcal {D}}}_{ \sigma }&{} \mathbb {A}^{{\mathcal {D}}}_{ \sigma ,\sigma ^*}\\ \mathbb {A}^{{\mathcal {D}}}_{ \sigma ,\sigma ^*}&{} \mathbb {A}^{{\mathcal {D}}}_{\sigma ^*}\end{bmatrix}, \end{aligned}$$
(A.1)
$$\begin{aligned} \mathbb {A}^{{\mathcal {D}},\Lambda }&= \dfrac{1}{4 m_{{\mathcal {D}}}} \begin{bmatrix} m_{ \sigma }^2 \Lambda {\mathbf{n}}_{ \sigma {\scriptscriptstyle K}}\cdot {\mathbf{n}}_{ \sigma {\scriptscriptstyle K}}\ &{} m_{ \sigma }m_{\sigma ^*}\Lambda {\mathbf{n}}_{ \sigma {\scriptscriptstyle K}}\cdot \mathbf{n}_{\sigma ^*{\scriptscriptstyle K^*}}\\ m_{ \sigma }m_{\sigma ^*}\Lambda {\mathbf{n}}_{ \sigma {\scriptscriptstyle K}}\cdot \mathbf{n}_{\sigma ^*{\scriptscriptstyle K^*}}&{} m_{\sigma ^*}^2 \; \Lambda \mathbf{n}_{\sigma ^*{\scriptscriptstyle K^*}}\cdot \mathbf{n}_{\sigma ^*{\scriptscriptstyle K^*}}\end{bmatrix} \nonumber \\&\;{=:} \begin{bmatrix} \mathbb {A}^{{\mathcal {D}},\Lambda }_{ \sigma }&{} \mathbb {A}^{{\mathcal {D}},\Lambda }_{ \sigma ,\sigma ^*}\\ \mathbb {A}^{{\mathcal {D}},\Lambda }_{ \sigma ,\sigma ^*}&{} \mathbb {A}^{{\mathcal {D}},\Lambda }_{\sigma ^*}\end{bmatrix}, \end{aligned}$$
(A.2)

and

$$\begin{aligned} \mathbb {B}^{{\mathcal {D}},\Lambda }= \begin{bmatrix} \left| \mathbb {A}^{{\mathcal {D}},\Lambda }_{ \sigma } \right| + \left| \mathbb {A}^{{\mathcal {D}},\Lambda }_{ \sigma ,\sigma ^*} \right| &{} 0 \\ 0 &{} \left| \mathbb {A}^{{\mathcal {D}},\Lambda }_{ \sigma ,\sigma ^*} \right| + \left| \mathbb {A}^{{\mathcal {D}},\Lambda }_{\sigma ^*} \right| \end{bmatrix}, \ \ \forall {\mathcal {D}}\in \mathfrak {D}. \end{aligned}$$
(A.3)

The following lemma claims a crucial property of the matrix \(\mathbb {A}^{{\mathcal {D}},\Lambda }\). In particular, it states that \(\mathbb {A}^{{\mathcal {D}},\Lambda }\) is positive definite.

Lemma A.1

[19] There exist some positive constants\(\lambda _0\)and\(\lambda _1\)depending only on the mesh regularity and on\(\underline{\Lambda }\), \(\overline{\Lambda }\)satisfying

$$\begin{aligned}&\mathbb {A}^{{\mathcal {D}},\Lambda }x \cdot x \le \mathbb {B}^{{\mathcal {D}},\Lambda }x \cdot x \le \lambda _1\mathbb {A}^{{\mathcal {D}},\Lambda }x \cdot x, \quad \forall x \in \mathbb {R}^2, \end{aligned}$$
(A.4)
$$\begin{aligned}&\lambda _0 \mathbb {A}^{{\mathcal {D}}}x \cdot x \le \mathbb {A}^{{\mathcal {D}},\Lambda }x \cdot x, \quad \forall x \in \mathbb {R}^2. \end{aligned}$$
(A.5)

Lemma A.2

Consider the following piecewise constant functions

$$\begin{aligned}&\overline{\xi }_{{{\mathcal {D}}}}^{n+1}{:}{=} \max \limits _{ M \in \mathcal {V}_{{\mathcal {D}}} } \left\{ \xi (u_M ^{n+1})\right\} , \ \ \ \ \ \ \ \underline{\xi }_{{{\mathcal {D}}}}^{n+1}{:}{=} \min \limits _{ M \in \mathcal {V}_{{\mathcal {D}}} } \left\{ \xi (u_M ^{n+1})\right\} ,\\&\overline{\xi }_{{\mathcal {T}, \delta t}}{_{{|{\mathcal {D}}\times (t^n,t^{n+1}]}}} {:}{=} \overline{\xi }_{{{\mathcal {D}}}}^{n+1}, \qquad \underline{\xi }_{{\mathcal {T}, \delta t}}{_{{|{\mathcal {D}}\times (t^n,t^{n+1}]}}}{:}{=} \underline{\xi }_{{{\mathcal {D}}}}^{n+1}, \end{aligned}$$

where we denote \(\mathcal {V}_{{\mathcal {D}}} = \{{ K}, { L}, { K^*},L^*\}.\) Then

$$\begin{aligned} \lim \limits _{h_{\mathfrak {D}}, \delta t\rightarrow 0} \left\| \overline{\xi }_{{\mathcal {T}, \delta t}}- \underline{\xi }_{{\mathcal {T}, \delta t}}\right\| _{{L^2(Q_T)}}= 0. \end{aligned}$$
(A.6)

Proof

The proof extends similar ideas provided in [20]. \(\square \)

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Quenjel, E.H., Saad, M., Ghilani, M. et al. Convergence of a positive nonlinear DDFV scheme for degenerate parabolic equations. Calcolo 57, 19 (2020). https://doi.org/10.1007/s10092-020-00367-5

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Keywords

  • Finite volume scheme
  • Positive
  • Degenerate parabolic equations
  • Convergence

Mathematics Subject Classification

  • 65M08
  • 65M12
  • 35K65