Numerical solution of the plane convection Darcy problem on a computer with distributed memory (Q2760717)

The following initial-boundary value problem is studied numerically for the equations of gravitational convection in the domain \(D\times (0,T)\): \(\Delta\psi = \theta_x\), \(\theta_t + \psi_y\theta_x - \psi_x \theta_y = \Delta\theta + \lambda\psi_x\), \(\psi(t,0,y) = \psi(t,a,y) = \psi(t,x,0) = \psi(t,x,b) = 0\), \(\theta(t,0,y) = \theta(t,a,y) = \theta(t,x,0) = \theta(t,x,b) =0\), \(\theta(0,x,y) = \theta_0(x,y)\), where \(D = (0,a)\times (0,b)\), \(\psi(x,y,t)\) is the stream function, \(\theta(t,x,y)\) denotes the temperature deviation from equilibrium state, and \(\lambda\) is Rayleigh number. The fact that this problem is solvable globally in time provides a reason to use Galerkin and Runge-Kutta methods for constructing an approximate solution. To realize the methods, the authors represent a parallel algorithm using a distributed memory computer nCube 2S. Moreover, the authors show how to distribute the data in parallel computations, and discuss the computational complexity of the algorithm. Numerical results are given for various distributions of parameter values. As a result of numerical simulation, the authors examine the appearance of periodic and chaotic modes.