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The random batch method provides an efficient algorithm for computing statistical properties of a canonical ensemble of interacting particles. In this work, we study the error estimates of the fully discrete random batch method, especially in terms of approximating the invariant distribution. The triangle inequality framework employed in this paper is a convenient approach to estimate the long-time sampling error of the numerical methods. Using the triangle inequality framework, we show that the long-time error of the discrete random batch method is |$O(\sqrt {\tau } + e^{-\lambda t})$|⁠ , where |$\tau $| is the time step and |$\lambda $| is the convergence rate, which does not depend on the time step |$\tau $| or the number of particles |$N$|⁠. Our results also apply to the McKean–Vlasov process, which is the mean-field limit of the interacting particle system as the number of particles |$N\rightarrow \infty $|⁠. [ABSTRACT FROM AUTHOR]

Compatible discrete operator schemes preserve basic properties of the continuous model at the discrete level. They combine discrete differential operators that discretize exactly topological laws and discrete Hodge operators that approximate constitutive relations. We devise and analyse two families of such schemes for the Stokes equations in curl formulation, with the pressure degrees of freedom located at either mesh vertices or cells. The schemes ensure local mass and momentum conservation. We prove discrete stability by establishing novel discrete Poincaré inequalities. Using commutators related to the consistency error, we derive error estimates with first-order convergence rates for smooth solutions. We analyse two strategies for discretizing the external load, so as to deliver tight error estimates when the external load has a large curl-free or divergence-free part. Finally, numerical results are presented on three-dimensional polyhedral meshes. [ABSTRACT FROM AUTHOR]

A set of accurate quadrature rules applicable to a class of integrable functions with isolated singularities is designed and analysed theoretically in one and two dimensions. These quadrature rules are based on the trapezoidal rule with corrected quadrature weights for points in the vicinity of the singularity. To compute the correction weights, small-size ill-conditioned systems have to be solved. The convergence of the correction weights is accelerated by the use of compactly supported functions that annihilate boundary errors. Convergence proofs with error estimates for the resulting quadrature rules are given in both one and two dimensions. The tabulated weights are specific for the singularities under consideration, but the methodology extends to a large class of functions with integrable isolated singularities. Furthermore, in one dimension we have obtained a closed form expression based on which the modified weights can be computed directly. [ABSTRACT FROM AUTHOR]