A Conservative Discontinuous Galerkin Solver for the Space Homogeneous Boltzmann Equation for Binary Interactions

Abstract

In the present work, we propose a deterministic numerical solver for the space homogeneous Boltzmann equation based on discontinuous Galerkin (DG) methods. Such an application has been rarely studied. The main goal of this manuscript is to generate a conservative solver for the collisional operator. As the key part, the weak form of the collision operator is approximated within subspaces of piecewise polynomials. In order to save the computational cost and to resolve loss of conservation laws due to numerical approximations, we propose the following combined procedures. First, the collision operator is projected onto a subspace of basis polynomials up to first order. Then, at every time step, a conservation routine is employed to enforce the preservation of desired moments (mass, momentum, and/or energy) by solving a constrained minimization problem with only linear complexity. The asymptotic error analysis shows the validity and guarantees the accuracy of these two procedures. The approximated collision integral is finally written as a quadratic form in a linear algebra setting. The theoretical number of operations for evaluating the complete set of the approximated collision matrix would be of order O(N 3) with N being the total number of freedom for d-dimensional velocity space. However, we have found and applied a "shifting symmetries" property in the collision weight matrix that consists of finding a minimal set of basis matrices that can exactly reconstruct the complete family of such a matrix. This procedure reduces the computation and storage of the collision matrix down to O(N 2). In addition, the matrix is highly sparse, yielding actual complexity O(N 2−1/d), with d being number of dimensions. Due to the locality of the DG schemes, the whole computing process is well performed with parallelization using hybrid OpenMP and message passing interface. The current work only considers the homogeneous Boltzmann equation with integrable angular cross sections under elastic and/or inelastic interaction laws. No transport is included in this manuscript. We only focus on the approximation of time dynamics for conservative binary collisions. The numerical results on two-dimensional and three-dimensional problems are provided.

Acceleration of Boltzmann Collision Integral Calculation Using Machine Learning

“…Many approaches have been proposed to enforce conservation laws in numerical evaluation of the collision integral, see, e.g., [17,39,40]. In this paper, we employ a modification of the Lagrangian multiplier method of [41,42]. The resulting postprocessing procedure is schematically described in Algorithm 2.…”

“…The procedure computes a corrected value of the collision integral that satisfies the discrete conservation laws up to roundoff errors while being as close as possible to the prediction. The difference from the approach of [42] is that values of the predicted collision integral that are small in magnitude are not affected by the procedure. Thus, the procedure avoids creation of small spurious values in the conservative collision integral at the domain boundaries.…”

Multi-fidelity methods for uncertainty propagation in kinetic equations

“…Regarding the error bound that we obtain using (104) with the values given by ( 103) le us observe that if, at each stage, we denote…”

“…The Boltzmann equation is used to model a very large number of different phenomena ranging from rarefied gas flows such as those found in hypersonic aerodynamics, gases in vacuum technologies, or fluids inside microelectromechanical devices [15,16], to the description of social and biological phenomena [64,74]. For these reasons, the development of efficient and accurate numerical methods for solving kinetic equations and in particular the Boltzmann equation has experienced great commitment in the past to which contributed many researchers working in different fields [9,13,31,42,43,65,70,72,104]. We refer to [20,22,71,72,88] for recent monographs, collections and surveys.…”

A conservative scheme for Vlasov Poisson Landau modeling collisional plasmas

“…Since the conservation routine (see Section 6) can force the conservation of any desired moments, we have to adjust it for the linear operator (88), which only conserves mass and energy. This is done by choosing a new 2 × 2N 3 constraint matrix by extracting only the first and fifth (in 3D case) rows of the full 5 × 2N 3 constraint matrix (35).…”