Efficient and accurate numerical approximation of the full Boltzmann equation has been a longstanding challenging problem in kinetic theory. This is mainly due to the high dimensionality of the problem and the complicated collision operator. In this work, we propose a highly efficient adaptive low rank method for the Boltzmann equation, concerning in particular the steady state computation. This method employs the fast Fourier spectral method (for the collision operator) and the dynamical low rank method to obtain computational efficiency. An adaptive strategy is introduced to incorporate the boundary information and control the computational rank in an appropriate way. Using a series of benchmark tests in 1D and 2D, we demonstrate the efficiency and accuracy of the proposed method in comparison to the full tensor grid approach.
翻译:整个波尔兹曼方程式的高效和准确数字近似是动能理论中长期存在的一个具有挑战性的问题,这主要是由于问题的高度维度和复杂的碰撞操作员。在这项工作中,我们提议对布尔兹曼方程式采用高度高效的适应性低级方法,特别是稳定状态计算。这种方法采用快速的傅里叶光谱方法(碰撞操作员)和动态低级方法,以获得计算效率。引入了适应性战略,以适当方式纳入边界信息和控制计算等级。我们使用1D和2D的一系列基准测试,展示了拟议方法与全长电网方法相比的效率和准确性。