A METHOD FOR FINDING A GENERALIZED SOLUTION WITH AN IMPLICIT DIFFERENCE SCHEME EXEMPLIFIED BY A QUASILINEAR TRANSPORT EQUATION

N. N. Bokov, E. G. Glinskikh
VANT. Ser. Metodiki i Programmy Chislennogo Resheniya Zadach Matematicheskoy Fiziki 1986. Вып.3. С. 61-67.

      A quasi linear transport equation is used to exemplify a method for computing large and small discontinuities in an implicit difference scheme. The continuous solution region uses an implicit difference scheme. For a large discontinuity the solution is fitted with respect to the Hugoniot condition. A small discontinuity requires that a continuity condition should be met. The complete system of difference equations is solved by three-point runs.
      The resulting difference scheme possesses a property to maintain small discontinuities in initial data, preserves the solution structure, reproduces satisfactorily the solution on a coarse spatial mesh, and converges to an exact solution as the timestep decreases.