142 research outputs found
The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schr\"odinger equation on a semi-infinite strip
We consider an initial-boundary value problem for a generalized 2D
time-dependent Schrodinger equation (with variable coefficients) on a
semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with
approximate or discrete transparent boundary conditions (TBCs), the Strang-type
splitting with respect to the potential is applied. For the resulting method,
the unconditional uniform in time -stability is proved. Due to the
splitting, an effective direct algorithm using FFT is developed now to
implement the method with the discrete TBC for general potential. Numerical
results on the tunnel effect for rectangular barriers are included together
with the detailed practical error analysis confirming nice properties of the
method.Comment: 23 pages, 6 figures, 4 table
On symmetric equilibrium of an isothermal gas with a free boundary and a body force
The equation of symmetric equilibrium of an isothermal gas with an
unknown boundary in the field of a body force is considered.
Conditions for solvability and insolvability of the problem as
well as for uniqueness and nonuniqueness of solutions are
presented. Examples of finite, countable, or continual sets of
solutions are constructed including equipotential ones. Static
stability of solutions is analyzed too
Stabilization for viscous compressible heat-conducting media equations with nonmonotone state functions
Verification of an entropy dissipative QGD-scheme for the 1D gas dynamics equations
An entropy dissipative spatial discretization has recently been constructed for the multidimensional gas dynamics equations based on their preliminary parabolic quasi-gasdynamic (QGD) regularization. In this paper, an explicit finite-difference scheme with such a discretization is verified on several versions of the 1D Riemann problem, both well-known in the literature and new. The scheme is compared with the previously constructed QGD-schemes and its merits are noticed. Practical convergence rates in the mesh L1-norm are computed. We also analyze the practical relevance in the nonlinear statement as the Mach number grows of recently derived necessary conditions for L2-dissipativity of the Cauchy problem for a linearized QGD-scheme
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