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An Ultra-Weak Discontinuous Galerkin Method with Implicit–Explicit Time-Marching for Generalized Stochastic KdV Equations

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Abstract

In this paper, an ultra-weak discontinuous Galerkin (DG) method is developed to solve the generalized stochastic Korteweg–de Vries (KdV) equations driven by a multiplicative temporal noise. This method is an extension of the DG method for purely hyperbolic equations and shares the advantage and flexibility of the DG method. Stability is analyzed for the general nonlinear equations. The ultra-weak DG method is shown to admit the optimal error of order \(k+1\) in the sense of the spatial \(L^2(0,2\pi )\)-norm for semi-linear stochastic equations, when polynomials of degree \(k\ge 2\) are used in the spatial discretization. A second order implicit–explicit derivative-free time discretization scheme is also proposed for the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples using Monte Carlo simulation are provided to illustrate the theoretical results.

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Acknowledgements

The authors would thank both reviewers and the editor for their careful reading and kind comments on the original manuscript.

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Authors and Affiliations

  1. Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Yunzhang Li & Shanjian Tang

  2. Division of Applied Mathematics, Brown University, Providene, RI, 02912, USA

    Chi-Wang Shu

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  1. Yunzhang Li
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  2. Chi-Wang Shu
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  3. Shanjian Tang
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Correspondence to Yunzhang Li.

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The work of the first and the third authors was supported by National Key R&D Program of China (Grant No. 2018YFA0703900) and National Natural Science Foundation of China (Grant No. 11631004). The work of the second author was supported by ARO Grant W911NF-16-1-0103 and NSF Grant DMS-1719410.

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Li, Y., Shu, CW. & Tang, S. An Ultra-Weak Discontinuous Galerkin Method with Implicit–Explicit Time-Marching for Generalized Stochastic KdV Equations. J Sci Comput 82, 61 (2020). https://doi.org/10.1007/s10915-020-01162-8

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  • DOI: https://doi.org/10.1007/s10915-020-01162-8

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