Chaoyu Quan

Chaoyu Quan, Benjamin Stamm, Yvon Maday

In this paper, a domain decomposition method for the Poisson-Boltzmann (PB) solvation model that is widely used in computational chemistry is proposed. This method, called ddLPB for short, solves the linear Poisson-Boltzmann (LPB) equation defined in $\mathbb R^3$ using the van der Waals cavity as the solute cavity. The Schwarz domain decomposition method is used to formulate local problems by decomposing the cavity into overlapping balls and only solving a set of coupled sub-equations in balls. A series of numerical experiments is presented to test the robustness and the efficiency of this method including the comparisons with some existing methods. We observe exponential convergence of the solvation energy with respect to the number of degrees of freedom which allows this method to reach the required level of accuracy when coupling with quantum mechanical descriptions of the solute.

Chaoyu Quan, Huaijin Wang, Xuping Wang et al.

This paper proposes a theoretical framework for establishing the energy dissipation of general implicit-explicit linear multistep methods (IMEX-LMMs) for gradient flows, by constructing a dissipative modified energy consisting of the original energy and a non-negative quadratic modification. We first test IMEX-LMMs with a simple multiplier, the first-order time difference of numerical solutions. Then, it is shown that the associated non-negative quadratic modification can be constructed if and only if two generating polynomials (corresponding to the LMM) are positive on $[-1,1]$. Based on this, the modified energy is proved to decay over time under a mild time-step restriction depending on the lower bounds of the associated generating polynomials. As a consequence, the energy dissipation of the well-known backward differentiation formula methods up to fifth order can be obtained straightforwardly. Furthermore, we construct for the first time (to the best of our knowledge) a sixth-order energy-dissipative IMEX-LMM and also prove the sixth-order barrier of energy-dissipative IMEX-LMMs when testing the simple multiplier. Some numerical experiments are conducted to verify our theoretical results.

Hong-lin Liao, Chaoyu Quan, Tao Tang et al.

The rigorous stability analysis of high-order implicit-explicit multistep (IEMS) methods for nonlinear parabolic equations by using discrete energy arguments is a long standing open issue due to their non-A-stable property. A novel semi-generating function approach combined with the global discrete energy analysis is suggested to the stability and convergence analysis of general IEMS methods for nonlinear parabolic equations. Inspired from the Grenander-Szegö theorem for the Toeplitz matrix, the semi-generating function approach is used to handle the three groups of discrete coefficients via three complex rational polynomials on the unit circle. A unified theoretical framework is then presented to establish the unconditional stability of IEMS methods if the minimum eigenvalue of composite convolution kernels for the implicit part is properly large and the spectral norm bound of composite convolution kernels for the explicit part is properly small. An indicator, called implicit-explicit controllability intensity, is then introduced to evaluate the degree of controllability of implicit part over explicit part. Some of existing IEMS methods, up to the fifth-order time accuracy, are revisited and compared by computing the associated implicit-explicit controllability intensities such that one can choose certain IEMS method or proper parameter to maintain the unconditional stability for a specific nonlinear parabolic model. We also propose a new parameterized class of IEMS methods, up to the eighth-order time accuracy, which satisfy the priori settings of our theory and have a large value of the implicit-explicit controllability intensity by choosing proper parameter so that they would be well suited for a wide class of nonlinear parabolic problems.

Yaru Liu, Chaoyu Quan, Dong Wang

This paper introduces a generalized matrix-valued Allen--Cahn model, where the unknown matrix-valued field belongs to $\mathbb{R}^{m_1\times m_2}$ with dimension $m_1\geq m_2$. By taking different values of $m_1$ and $m_2$, this model covers the classical scalar-valued, vector-valued, and square-matrix-valued Allen--Cahn equations. At the continuous level, the proposed model is proven to admit a unique solution satisfying the maximum bound principle (MBP) and the energy dissipation law. At the discrete level, a class of arbitrarily high-order exponential time differencing Runge-Kutta (ETDRK) schemes is investigated that preserve the MBP unconditionally. Moreover, we prove that the first- and second-order ETDRK schemes satisfy the discrete energy dissipation unconditionally, while third- and higher-order schemes preserve the discrete energy dissipation under suitable time-step constraints. The proof of sharp convergence order in time is provided. Numerical experiments are carried out to confirm our theoretical results.