An efficient finite-difference method for solving Smoluchowski-type kinetic equations of aggregation with three-body collisions

Рассмотрена модель агрегационных процессов для класса уравнений типа уравнений Смолуховского, допускающих тройные взаимодействия агрегатов. Предложен численный метод быстрого решения задачи Коши для указанной системы уравнений, позволяющий снизить алгоритмическую сложность $O \l(N^{3}\r)$ шага выполнения разностной схемы предиктор-корректор до $O (RN\log N)$ без потери точности, где $N$ задает количество используемых уравнений, а $R$ определяет ранг массивов кинетических коэффициентов. Эффективность и точность нового численного метода продемонстрированы для модельных задач агрегационной кинетики. We consider a model of aggregation processes for the Smoluchowski-type kinetic equations with three-body collisions of particles. We propose a numerical method for the fast solving of Cauchy problems for the corresponding systems of equations. The proposed method allows one to reduce the step complexity $O \l(N^{3}\r)$ of the finite-difference predictor-corrector scheme to $O (RN\log N)$ without loss of accuracy. Here the parameter $N$ specifies the number of considered equations and $R$ is the rank of kinetic coefficient arrays. The efficiency and accuracy of the proposed numerical method are demonstrated for model problems of aggregation kinetics.

21st-century humanities: Art, complexity, and interdisciplinarity

This article contends that the evolution toward interdisciplinary collaboration that we are witnessing in the sciences must also occur in the humanities to ensure their very survival. That is, humanists must be open to working with scientists and social scientists interested in similar research questions and vice versa. Digital humanities is a positive first step. Complexity science should be the next step. Even though much of the ground-breaking work in complexity science has been done in the natural sciences and mathematics, it can, if critically adapted, provide the needed metaphor for a broad integration of disciplines, humanistic and otherwise. Given its almost a-disciplinary nature, a complexity approach to the research problems in the humanities necessarily breaks down silos. Moreover, it can restore and reframe the seamless intellectual fabric sought by researchers before the atomization of the various disciplines in the nineteenthcentury academy.