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The pointwise error estimates of two energy-preserving fourth-order compact schemes for viscous Burgers’ equation

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Abstract

A novel fourth-order three-point compact operator for the nonlinear convection term uux is provided in this paper. The operator makes the numerical analysis of higher-order difference schemes become possible for a wide class of nonlinear evolutionary equations under the unified framework. We take the classical viscous Burgers’ equation as an example and establish a new conservative fourth-order implicit compact difference scheme based on the method of order reduction. A detailed theoretical analysis is carried out by the discrete energy argument and mathematical induction. It is rigorously proved that the difference scheme is conservative, uniquely solvable, stable, and unconditionally convergent in discrete \(L^{\infty }\)-norm. The convergence order is two in time and four in space, respectively. Furthermore, we derive a three-level linearized compact difference scheme for viscous Burgers’ equation based on the proposed operator. All numerically theoretical results similar to that of the nonlinear numerical scheme are inherited completely; meanwhile, it is more time saving. Applying the compact operator to other more complex and higher-order nonlinear evolutionary equations is feasible, including Benjamin-Bona-Mahony-Burgers’ equation, Korteweg-de Vries equation, Kuramoto-Sivashinsky equation, and classification to name a few. Numerical results demonstrate that the presented schemes for Burgers’ equation can achieve second-order accuracy in time and fourth-order accuracy in space in \(L^{\infty }\)-norm.

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Acknowledgements

We would like to thank Jan S. Hesthaven in École Polytechnique Fédérale de Lausanne (EPFL) for valuable comments and suggestions.

Funding

Sun was supported by the Natural Science Foundation of China under Grant 11671081. Zhang was supported in part by project funded by China Postdoctoral Science Foundation under Grant 2018M642131 when he studied in Southeast University, in part by the Natural Science Foundation of China under Grant 11501514, and in part by the Visiting Scholar Program of China Scholarship Council under Grant 201908330528.

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, China

    Xuping Wang, Qifeng Zhang & Zhi-zhong Sun

  2. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Qifeng Zhang

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  1. Xuping Wang
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Correspondence to Qifeng Zhang.

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Communicated by: Jan Hesthaven

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Appendix. Direct proof of the convergence of the difference scheme in L 2-norm

Appendix. Direct proof of the convergence of the difference scheme in L 2-norm

The result and direct proof of L2-norm error estimate for the convergence of the difference scheme (3.14)–(3.18) are listed as follows.

Theorem A.1

Let {u(x,t), v(x,t)} be the solution of (3.1)–(3.4) and \(\{{u_{i}^{k}},\ {v_{i}^{k}} | 0 \leqslant i \leqslant M, 0 \leqslant k \leqslant N\}\) be the solution of the difference scheme (3.14)–(3.18). Denote

$${e_{i}^{k}} = {U_{i}^{k}}-{u_{i}^{k}}, \quad {f_{i}^{k}} = {V_{i}^{k}}-{v_{i}^{k}}, \quad 0 \leqslant i \leqslant M, 0 \leqslant k \leqslant N,$$

and

$$ c_{13}= \frac12\left( \frac{2{c_{3}^{2}}}{\nu } + \nu + \frac{c_{3}}2\right), \quad c_{14} = \left( \nu + \frac1{2\nu}\right) {c_{1}^{2}}L, \quad c_{15} = \exp\left\{3c_{13}T\right\} \cdot \sqrt{\frac{c_{14}}{2c_{13}}}. $$

If \(2c_{13}\tau \leqslant 1/3\), then we have:

$$ \|e^{k}\| \leqslant c_{15}(\tau^{2}+h^{4}),\quad 0 \leqslant k \leqslant N. $$
(A.1)

Proof

It is easy to know that (A.1) holds for k = 0.

Taking an inner product of (3.41) with \(e^{k+\frac {1}{2}}\), we have:

$$ \begin{array}{@{}rcl@{}} && \left( \delta_{t}e^{k+\frac{1}{2}},e^{k+\frac{1}{2}}\right) + \left( \psi(U^{k+\frac{1}{2}},U^{k+\frac{1}{2}}) - \psi(u^{k+\frac{1}{2}},u^{k+\frac{1}{2}}),e^{k+\frac{1}{2}} \right) \\ &&-\frac{h^{2}}{2}\left( \psi(V^{k+\frac{1}{2}},U^{k+\frac{1}{2}})-\psi(v^{k+\frac{1}{2}},u^{k+\frac{1}{2}}),e^{k+\frac{1}{2}}\right)-\nu \left( f^{k+\frac{1}{2}},e^{k+\frac{1}{2}}\right) = \left( P^{k+\frac{1}{2}},e^{k+\frac{1}{2}}\right), \\ && \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 0 \leqslant k \leqslant N-1. \end{array} $$
(A.2)

Using Lemma 2.1, Cauchy-Schwartz inequality, and inequality (2.4), we have:

$$ \begin{array}{@{}rcl@{}} && \left( f^{k+\frac{1}{2}},e^{k+\frac{1}{2}}\right) \\ &\leqslant & -|e^{k+\frac{1}{2}}|_{1}^{2} -\frac{h^{2}}{18}\|f^{k+\frac{1}{2}}\|^{2} + \frac{h^{2}}{12} \left( f^{k+\frac{1}{2}},Q^{k+\frac{1}{2}}\right) + \left( Q^{k+\frac{1}{2}},e^{k+\frac{1}{2}}\right) \\ &\leqslant & -\frac{h^{2}}{18}\|f^{k+\frac{1}{2}}\|^{2} + \frac{h^{2}}6\left( \frac19\|f^{k+\frac12}\|^{2} + \frac9{16}\|Q^{k+\frac{1}{2}}\|^{2}\right) + \left( \frac12\|Q^{k+\frac{1}{2}}\|^{2} + \frac12\|e^{k+\frac{1}{2}}\|^{2}\right) \\ &= & - \frac{h^{2}}{27}\|f^{k+\frac12}\|^{2} + \left( \frac{3h^{2}}{32} + \frac12\right)\|Q^{k+\frac{1}{2}}\|^{2} + \frac12\|e^{k+\frac{1}{2}}\|^{2}. \end{array} $$
(A.3)

By Lemmas 2.2, 2.3 and (3.39), computing arrives at:

$$ \begin{array}{@{}rcl@{}} && - \left( \psi(U^{k+\frac{1}{2}},U^{k+\frac{1}{2}}) - \psi(u^{k+\frac{1}{2}},u^{k+\frac{1}{2}}),e^{k+\frac{1}{2}}\right)\\ &=& -\left( \psi(e^{k+\frac{1}{2}},U^{k+\frac{1}{2}}),e^{k+\frac{1}{2}}\right)\\ &=& - \frac{h}{3} \sum\limits_{i=1}^{M-1} \left[e_{i}^{k+\frac{1}{2}}{\varDelta}_{x} U_{i}^{k+\frac{1}{2}}+ {\varDelta}_{x}(e^{k+\frac{1}{2}}U^{k+\frac{1}{2}})_{i} \right]e_{i}^{k+\frac{1}{2}}\\ &=& - \frac{h}{3} \sum\limits_{i=1}^{M-1} (e_{i}^{k+\frac{1}{2}})^{2} \cdot {\varDelta}_{x} U_{i}^{k+\frac{1}{2}}- \frac{h}{6} \sum\limits_{i=1}^{M-1} \frac{U_{i+1}^{k+\frac{1}{2}}-U_{i}^{k+\frac{1}{2}}}{h} \cdot e_{i}^{k+\frac{1}{2}}e_{i+1}^{k+\frac{1}{2}}\\ &\leqslant & \frac{c_{3}}{2}\|e^{k+\frac{1}{2}}\|^{2} \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} && \frac{h^{2}}{2}\left( \psi(V^{k+\frac{1}{2}},U^{k+\frac{1}{2}})-\psi(v^{k+\frac{1}{2}},u^{k+\frac{1}{2}}),e^{k+\frac{1}{2}}\right) \\ &= & \frac{h^{2}}2\left( \psi(f^{k+\frac{1}{2}},U^{k+\frac{1}{2}}),e^{k+\frac{1}{2}}\right) \\ &= & \frac{h^{3}}6 \sum\limits_{i=1}^{M-1} \left[f_{i}^{k+\frac{1}{2}}{\varDelta}_{x}U_{i}^{k+\frac{1}{2}} +{\varDelta}_{x}(f^{k+\frac{1}{2}}U^{k+\frac{1}{2}})_{i}\right]e_{i}^{k+\frac{1}{2}} \\ &=& \frac{h^{3}}6 \sum\limits_{i=1}^{M-1} f_{i}^{k+\frac{1}{2}}e_{i}^{k+\frac{1}{2}}{\varDelta}_{x}U_{i}^{k+\frac{1}{2}} -\frac{h^{3}}6\sum\limits_{i=1}^{M-1}f_{i}^{k+\frac{1}{2}}U_{i}^{k+\frac{1}{2}}{\varDelta}_{x}e_{i}^{k+\frac{1}{2}} \\ &\leqslant & \frac{c_{3}h^{2}}{6} \|f^{k+\frac{1}{2}}\|\cdot\|e^{k+\frac{1}{2}}\| + \frac{c_{3}h^{2}}{6}\|f^{k+\frac{1}{2}}\|\cdot |e^{k+\frac{1}{2}}|_{1} \\ &\leqslant & \frac{h^{2}}6\left( \frac\nu9\|f^{k+\frac12}\|^{2} + \frac{9{c_{3}^{2}}}{4\nu}\|e^{k+\frac12}\|^{2}\right) + \frac{h^{2}}6\left( \frac\nu9\|f^{k+\frac12}\|^{2} + \frac{9{c_{3}^{2}}}{4\nu}|e^{k+\frac12}|_{1}^{2}\right) \\ &\leqslant & \frac{\nu h^{2}}{27}\|f^{k+\frac12}\|^{2} + \left( \frac{3{c_{3}^{2}}h^{2}}{8\nu} + \frac{3{c_{3}^{2}}}{2\nu }\right)\|e^{k+\frac12}\|^{2}. \end{array} $$
(A.4)

Substituting (A.3)–(A.4) into (A.2) and combining with (3.8) and (3.9), we can obtain:

$$ \begin{array}{@{}rcl@{}} & & \frac{1}{2\tau}(\|e^{k+1}\|^{2}-\|e^{k}\|^{2})+ \frac{\nu h^{2}}{27}\|f^{k+\frac{1}{2}}\|^{2} \\ &\leqslant & \left( \frac{3\nu h^{2}}{32} + \frac\nu2\right)\|Q^{k+\frac{1}{2}}\|^{2} + \frac\nu2\|e^{k+\frac{1}{2}}\|^{2} + \frac{c_{3}}2\|e^{k+\frac12}\|^{2} \\ & &+ \frac{\nu h^{2}}{27}\|f^{k+\frac12}\|^{2} + \left( \frac{3{c_{3}^{2}}h^{2}}{8\nu} + \frac{3{c_{3}^{2}}}{2\nu }\right)\|e^{k+\frac12}\|^{2}+(P^{k+\frac{1}{2}},e^{k+\frac{1}{2}}), \quad 0 \leqslant k \leqslant N-1 \end{array} $$

or

$$ \begin{array}{@{}rcl@{}} && \frac{1}{2\tau}\left( \|e^{k+1}\|^{2}-\|e^{k}\|^{2}\right) \\ &\leqslant & \nu\|Q^{k+\frac{1}{2}}\|^{2} + \left( \frac{2{c_{3}^{2}}}{\nu } + \nu + \frac{c_{3}}2\right)\|e^{k+\frac12}\|^{2}+ \frac1{2\nu}\|P^{k+\frac{1}{2}}\|^{2} \\ &\leqslant & c_{13}\left( \|e^{k+1}\|^{2} + \|e^{k}\|^{2}\right) + c_{14}(\tau^{2}+h^{4})^{2}, \quad 0 \leqslant k \leqslant N-1. \end{array} $$

Then we have

$$(1-2c_{13}\tau)\|e^{k+1}\|^{2} \leqslant (1+2c_{13}\tau) \|e^{k}\|^{2} + 2 c_{14}\tau(\tau^{2}+h^{4})^{2}, \quad 0 \leqslant k \leqslant N-1.$$

When \(2c_{13}\tau \leqslant \frac {1}{3}\), we have

$$\|e^{k+1}\|^{2} \leqslant (1+6c_{13}\tau) \|e^{k}\|^{2} + 3c_{14}\tau(\tau^{2}+h^{4})^{2}, \quad 0 \leqslant k \leqslant N-1.$$

Using Gronwall inequality, equations (3.43) and (3.44), we can obtain

$$\|e^{k+1}\|^{2} \leqslant \exp\left\{6c_{13}k\tau\right\} \cdot \frac{c_{14}}{2c_{13}} \cdot (\tau^{2}+h^{4})^{2} \leqslant c_{15}^{2}(\tau^{2}+h^{4})^{2}, \quad 0 \leqslant k \leqslant N-1.$$

This completes the proof. □

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Wang, X., Zhang, Q. & Sun, Zz. The pointwise error estimates of two energy-preserving fourth-order compact schemes for viscous Burgers’ equation. Adv Comput Math 47, 23 (2021). https://doi.org/10.1007/s10444-021-09848-9

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