2023 , Volume 28, № 3, p.101-116

Purpose. The purpose of the study is the development and comparison of two numerical semi-Lagrangian methods with fulfillment of the conservation law at a discrete level. The approach is applied for the transport equation in the symmetric form, reflecting the law of conservation for the square of the transferred substance. The article presents the Euler – Lagrangian method, built on a rectangular difference grid that uses local values of characteristics to calculate the coefficients of difference equations. Lagrangian – Euler method is built on a spatial non-uniform grid obtained by crossing the characteristic trajectories of the equation with lines in time. Methodology. The integro-interpolation method is applied to derive approximations for the differential operator which allowed obtaining simple formulas connecting values of the grid function at the neighboring layers in time. Numerical calculations of characteristic trajectories are held by the Euler method or the Runge – Kutta method of the second order, depending on the required accuracy. Findings. Numerical methods with the mentioned properties are developed and numerically confirmed, convergence and discrete conservation laws for them are mathematically proved. The first order convergence for both time and space is proved for the Euler – Lagrange method. The second order convergence also in time and space is proved for the Lagrange – Euler method. Originality/value. The Euler – Lagrange and Lagrange – Euler methods for the numerical solution of the convection equation are developed. These methods induce differential conservation law at discrete level. The first and the second order of convergence correspondingly are mathematically proved for them. The Lagrange – Euler method has showed two improved aspects: firstly, it has greater order of convergence than the Euler – Lagrange one and secondly, it allows solving problems with the discontinuous solutions without smoothing them