In this paper, we develop a set of efficient methods to compute stationary states of the spherical Landau-Brazovskii (LB) model in a discretization-then-optimization way. First, we discretize the spherical LB energy functional into a finite-dimensional energy function by the spherical harmonic expansion. Then five optimization methods are developed to compute stationary states of the discretized energy function, including the accelerated adaptive Bregman proximal gradient, Nesterov, adaptive Nesterov, adaptive nonlinear conjugate gradient and adaptive gradient descent methods. To speed up the convergence, we propose a principal mode analysis (PMA) method to estimate good initial configurations and sphere radius. The PMA method also reveals the relationship between the optimal sphere radius and the dominant degree of spherical harmonics. Numerical experiments show that our approaches significantly reduce the number of iterations and the computational time.

The maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations. To propose MBP-preserving schemes with high spatial accuracy, in the first part of this series, we developed a class of time semidiscrete stochastic Runge-Kutta (SRK) methods for semilinear parabolic equations, and constructed the first- and second-order fully discrete MBP-preserving SRK schemes. In this paper, to develop higher order fully discrete MBP-preserving SRK schemes with spectral accuracy in space, we use the Sinc quadrature rule to approximate the conditional expectations in the time semidiscrete SRK methods and propose a class of fully discrete MBP-preserving SRK schemes with up to fourth-order accuracy in time for semilinear equations. Based on the property of the Sinc quadrature rule, we theoretically prove that the proposed fully discrete SRK schemes preserve the MBP and can achieve an exponential order convergence rate in space. In addition, we reveal that the conditional expectation with respect to the Bronwian motion in the time semidiscrete SRK method is essentially equivalent to the exponential Laplacian operator under the periodic boundary condition. Ample numerical experiments are also performed to demonstrate our theoretical results and to show the exponential order convergence rate in space of the proposed schemes.

In this work, we consider the dynamic framework of the ocean-atmosphere (O-A) coupled model with physical boundary conditions at the ocean-atmosphere interface, and this coupled model can be viewed as atmosphere general circulation model coupled with ocean general circulation model. As the initial data and boundary conditions are assumed to meet certain assumptions, by taking advantage of energy estimates method and compactness arguments, we addressed the existence and stability of global weak solutions, the existence and uniqueness of global strong solution to the O-A coupled model.