SM stability for time-dependent problems

When time-dependent problems are solved approximately, the accuracy can be improved in various ways. In the case of two-level schemes (the solution at two adjacent time levels is involved), polynomial approximations of the scheme operators on the solutions are used explicitly or implicitly. The most popular representatives of such schemes are Runge-Kutta methods [7,8], which are widely used in modern computations. The main feature of the multilevel schemes (multistep methods) manifests itself in the approximation of time derivatives with a higher accuracy on a multipoint stencil. A characteristic example is provided by multistep methods based on backward numerical differentiation [9].

Various classes of stable finite difference schemes can be constructed to obtain a numerical solution [10,11]. It is important to select among all stable schemes such a scheme that is optimal in terms of certain additional criteria. In the theory of finite difference schemes, there is the class of asymptotically stable schemes (see [12,13]) that ensure the correct long-time behavior of the approximate solution. In the theory of numerical methods for ODEs (see [7,9]), the concept of L-stability is used, which reflects the long-time asymptotic behavior of the approximate solution from a different point of view.

In [14] there are considered properties of two-level difference schemes of high order approximation for the approximate solution of the Cauchy problem for evolutionary equations with self-adjoint operators. The simplest boundary value problem for the one-dimensional parabolic equation serves as a basic problem. The concept of SM stability (Spectral Mimetic stability) of a difference scheme is introduced. This property is connected with the behavior of individual harmonics of the approximate solutions.

In this paper, we continue to study the SM properties of difference schemes for the approximate solution of unsteady problems of mathematical physics. For the model boundary value problem for one-dimensional parabolic equation there are considered the spectral characteristics of approximations in space and in time. In particular, good approximation properties are observed for the convection operator of third order in space. Features two-level schemes of higher order of approximation in time, based on the Padé approximation, are considered for solving problems of mathematical physics with symmetric and skew-symmetric operators.

We consider finite-dimensional real Hilbert space H, where the scalar product and the norm is (•, •) and • , respectively. Let u(t) (0 ≤ t ≤ T > 0) is defined as the solution of the Cauchy problem for evolutionary equation of first order:

The right part f (t) ∈ H of equation ( 1) is given and Λ is a linear non-negative, in general, nonself-adjoint operator from H in H (Λ = Λ(t) ≥ 0) depending on t.

For problem (1), ( 2) the estimate of stability is easily established. Taking into account the skew-symmetric property of operator Λ, we have the equality

we obtain a simple estimate of stability for the solution of ( 1), ( 2) with respect to the initial data and right hand side:

We would like to preserve these properties of the differential problem after transition to a discrete analogue of problem ( 1), ( 2). The main attention in our discussion is given to unsteady boundary value problem for partial differential equations. In this context, we can associate the Cauchy problem (1), ( 2) with application of the method of lines (approximation in space). Given the importance for applications, we will conduct our consideration on an example of a boundary value problem for the one-dimensional parabolic equation of second order. Let a sufficiently smooth function u(x, t) satisfies the equation

and the initial conditions

The periodicity of the spatial variable is assumed:

We associate operator L with the convection-diffusion equation, defining

for some constant 0 ≤ χ ≤ 1. Here the operators of convective and diffusive transport are defined according

If χ = 1 equation ( 4) is the convection transport equation whereas if χ = 0 it is the diffusion equation.

The discrete problem should inherit the main properties of the differential problem. In model problem ( 4)-( 6) the skew-symmetric property of operator C as well as the self-adjoint and non-negative properties of o

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