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Abstract: A numerical method devised for the solution of the Vlasov–Poisson coupled equations in three-dimensional phase space (radial distance, absolute value of the velocity and µ ≡ cos (v, r) is described. The method is based on the Lax–Wendroff integration scheme with a specially adapted Shuman filter. The method provides a second order accurate and conservative (in mass) direct integration of the above coupled system without making additional assumptions, like expansion into moments, local equilibrium etc. The method is used to analyse the non-linear dynamic stability of spherical self-gravitating systems. We conclude that some Newtonian stellar clusters which are found to be thermodynamically unstable are dynamically stable. The method allows us to study the details of the relaxation towards a steady state. We find that the long spatial wavelengths relax first and the relaxation proceeds towards shorter wavelengths.