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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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
Keywords
- Ultra-weak discontinuous Galerkin method
- Generalized stochastic KdV equations
- Multiplicative stochastic noise
- Stability analysis
- Error estimates
- Implicit–explicit time discretization