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The resulting system of radial equations containing the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method.

The resulting system of these equations containing the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method.

The resulting system of radial equations which contains the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite-element method.

Even singularly perturbed boundary value problems with or without first derivative terms are solved.

The other source terms are solved using an explicit Euler approximation.

The stiff source terms are solved by an implicit trapezoidal method of integration.

Second, stiff relaxation terms are solved with a Newton method that also guarantees positivity and robustness.

The elliptic problems coming from the viscous terms are solved by means of the Chebyshev acceleration method.

The wall-normal viscous and diffusive terms are solved implicitly while the convective terms and the remaining viscous and diffusive terms are treated explicitly.

The diffusion terms are solved by the explicit second order orthogonal Chebyshev method (ROCK2), while the stiff reaction terms (solved implicitly) and the advection and noise terms (solved explicitly) are integrated in the algorithm as finishing procedures.

Moreover, the errors in quantities of interest are smaller than those from the Strang splitting scheme indicating the accuracy gain when the reaction and transport terms are solved coupled.