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High-Order Semi-Lagrangian WENO Schemes Based on Non-polynomial Space for the Vlasov Equation

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Abstract

In this paper, we present a semi-Lagrangian (SL) method based on a non-polynomial function space for solving the Vlasov equation. We find that a non-polynomial function based scheme is suitable to the specifics of the target problems. To address issues that arise in phase space models of plasma problems, we develop a weighted essentially non-oscillatory (WENO) scheme using trigonometric polynomials. In particular, the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities. Moreover, to obtain a high-order of accuracy in not only space but also time, it is proposed to apply a high-order splitting scheme in time. We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system. Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions. A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method. In 6D, this would represent a significant savings.

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Acknowledgements

We would like to thank AFOSR and NSF for their support of this work under grants FA9550-19-1-0281 and FA9550-17-1-0394 and NSF grant DMS 191218.

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Authors and Affiliations

  1. Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Andrew Christlieb, Matthew Link & Hyoseon Yang

  2. Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, Jiangsu, China

    Ruimeng Chang

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Correspondence to Hyoseon Yang.

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Appendices

Appendix A Formulation of \(\mathbf {C}\) and \(\mathbf {c}_k\)

The numerical flux \({\hat{f}}_{j-\frac{1}{2}}\) in (25) based on the function space \(\varGamma _5 = \mathrm{span} \{\cos {x},\sin { x}, x^2, x^3, x^4\}\) is constructed by using the coefficients

$$\begin{aligned} \mathbf {C} = \left[ \frac{550}{16\,403}, -\frac{157}{724}, \frac{1\,587}{2\,057}, \frac{281}{599}, -\frac{155}{3\,016} \right] ^{\text T}. \end{aligned}$$

The local numerical flux \({\hat{f}}_{j-\frac{1}{2}}^k\) in (26) can be obtained based on algebraic function space \(\varPi _3 =\mathrm{span}\{1, x, x^2\}\) with the constants

$$\begin{aligned} \mathbf {c}^0&= \left[ \frac{1}{3} ,-\frac{7}{6}, \frac{11}{6} \right] ^{\text T}, \\ \mathbf {c}^1&= \left[ -\frac{1}{6} , \frac{5}{6}, \frac{1}{3} \right] ^{\text T}, \\ \mathbf {c}^2&= \left[ \frac{1}{3} , \frac{5}{6},-\frac{1}{6} \right] ^{\text T}, \end{aligned}$$

or based on the function space \(\mathrm{span} \{1, \sin {x},\cos { x}\}\) with the coefficients

$$\begin{aligned} \mathbf {c}^0&= \left[ \frac{109}{280 } , -\frac{478}{471 }, \frac{ 1\,411}{868 } \right] ^{\text T}, \\ \mathbf {c}^1&= \left[ -\frac{826}{4\,031} , \frac{1\,973}{2\,419}, \frac{ 109}{280 } \right] ^{\text T}, \\ \mathbf {c}^2&= \left[ \frac{109}{280 } , \frac{1\,973}{2\,419}, -\frac{ 826}{4\,031 } \right] ^{\text T}. \end{aligned}$$

Appendix B Proof of Proposition 2

Assuming f smooth enough around the stencil, we can represent the smoothness indicators (28) using Taylor expansion as

$$\begin{aligned} \begin{aligned} \beta _0&= \left( \Delta x f'_{j-\frac{1}{2}} -\frac{23}{24} \Delta x^3 f'''_{j-\frac{1}{2}} + {{\mathcal {O}}}(\Delta x^4) \right) ^2 + \left( \Delta x^2 f''_{j-\frac{1}{2}} -\frac{3}{2} \Delta x^3 f'''_{j-\frac{1}{2}} + {{\mathcal {O}}}(\Delta x^4) \right) ^2, \\ \beta _1&= \left( \Delta x f'_{j-\frac{1}{2}} +\frac{1}{24} \Delta x^3 f'''_{j-\frac{1}{2}} + {{\mathcal {O}}}(\Delta x^4) \right) ^2 + \left( \Delta x^2 f''_{j-\frac{1}{2}} -\frac{1}{2} \Delta x^3 f'''_{j-\frac{1}{2}} + {{\mathcal {O}}}(\Delta x^4) \right) ^2, \\ \beta _2&= \left( \Delta x f'_{j-\frac{1}{2}} +\frac{1}{24} \Delta x^3 f'''_{j-\frac{1}{2}} + {{\mathcal {O}}}(\Delta x^4) \right) ^2 + \left( \Delta x^2 f''_{j-\frac{1}{2}} +\frac{1}{2} \Delta x^3 f'''_{j-\frac{1}{2}} + {{\mathcal {O}}}(\Delta x^4) \right) ^2. \end{aligned} \end{aligned}$$

Then for each \(k=0,1,2\), we can obtain

$$\begin{aligned} 1+\frac{\tau }{(\epsilon +\beta _k)^2}&= 1+\frac{(\beta _2 - \beta _0)^2}{(\epsilon +\beta _k)^2} \\&= 1+\frac{\left( 2\Delta x^4 f'_{j-\frac{1}{2}} f'''_{j-\frac{1}{2}} + {{\mathcal {O}}}(\Delta x^5) \right) ^2}{(\epsilon +\beta _k)^2} \\&= 1+ C \Delta x^4 + {{\mathcal {O}}}(\Delta x^5) \end{aligned}$$

with a constant C, by choosing \(\epsilon = \Delta x^2\), so that it is straightforward by the definition of \(\omega _k\) in (29).

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Christlieb, A., Link, M., Yang, H. et al. High-Order Semi-Lagrangian WENO Schemes Based on Non-polynomial Space for the Vlasov Equation. Commun. Appl. Math. Comput. 5, 116–142 (2023). https://doi.org/10.1007/s42967-021-00150-5

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