Numerical projection of a probability distribution function to enforce linear equality constraints

Abstract

Conservation laws are fundamental in describing physical phenomena, particularly in plasma physics where collisions between charged particles must conserve mass, momentum, and energy. The Fokker-Planck equation, which inherently satisfies these conservation constraints through its mathematical structure, is widely used in this field. However, the continuous-space formulation often fails to preserve these constraints when discretized for numerical analysis without careful consideration. Broadly, two approaches have emerged to ensure numerical conservation. One approach focuses on carefully designing numerical discretization schemes that inherently satisfy conservation properties. The other approach, in contrast, applies more general discretization methods without explicit consideration of conservation, followed by a posterior correction to enforce conservation on the solution. Following the latter approach, this study proposes a projection-based correction method that leverages the fact that conserved quantities such as mass, momentum, and energy in the numerically discretized form are evaluated by linearly multiplying the probability distribution function. By spanning the null vectors in multidimensional space of the distribution function, we project a solution vector onto a hyperplane that satisfies conservation of the conserved quantities. This operation minimizes change of the solution vector in 𝐿2 sense and consequently is expected to impact less change on main physics operation in turn. We apply this method to single and two-species collision test cases, demonstrating its stability and machine-level accuracy compared to existing free-parameter-based correction methods. Our approach offers significant advantages in computational efficiency and implementation simplicity, requiring only basic algebraic operations without complex differential terms or iterative calculations. Its versatility extends beyond continuum methods using probability distribution functions to particle simulations for weight adjustment, and it is applicable to a wide range of problems characterized by linear equality constraints.