Track method for the calculation of plasma heating by charged thermonuclear reaction products for axisymmetric flows

Integral formulas for the three-dimensional case that give the plasma heating rate per unit volume are obtained using the track method and by integrating the well-known Cauchy problem for the steady-state homogeneous kinetic equation in the Fokker-Planck approximation in the absence of diffusion of the distribution function in the velocity space and under the condition that the velocity of the produced particles is independent on the direction of their escape. It is shown that both integral formulas are equivalent and, in the case of space homogeneous coefficients, turn into the model of local plasma heating away from the domain boundary. In addition to the known direct track method, the inverse method based on the approximation of the integral formula is developed. It is shown that the accuracy of the direct method is significantly decreased in the vicinity of the symmetry axis for not very fine angular grids. In the inverse method, the accuracy is not lost. It is shown that the computational cost of the inverse method can be significantly reduced without the considerable reduction of the computation accuracy.