INTRODUCTION

The hyperbolic quasi-gasdynamic system is a specially perturbed system of gasdynamic equations in which the terms containing second derivatives with respect to the space and time variables are multiplied by a small parameter \(\tau >0\). This system is used for constructing a family of three-level and two-level vector difference schemes for the numerical solution of various problems in gas dynamics [1]. Note that parabolic quasi-gasdynamic systems are presented in [2, 3]. The papers [4, 5] establish a number of mathematical properties of the quasi-gasdynamic system, including time-uniform estimates for the corresponding system linearized on a constant solution. A second-order hyperbolic perturbation with the parameter \(\tau \) of the parabolic initial–boundary value problem without convective terms [6, 7] and with them [8] was also studied, and the stability of the corresponding three-level weighted and two-level vector numerical methods was analyzed [9]. The time-uniform stability of implicit three-level weighted and two-level vector difference schemes for the linearized quasi-gasdynamic system was proved in [10].

In practice, explicit difference schemes for the quasi-gasdynamic system are of most interest. However, the linearizations of these schemes have not yet been proved to be stable uniformly in \(\tau \). So far, the energy method has not been successfully applied to prove this by analogy with [4, 10], because the linearized schemes have a rather complex structure; in particular, they involve approximations to terms containing first and second derivatives with respect to the spatial and temporal variables, the second derivatives being multiplied by the parameter \(\tau \).

The present paper is intended to make some progress in this direction. To this end, instead of the quasi-gasdynamic system, we consider the simplified case of one second-order hyperbolic equation with variable coefficients that is a perturbation of the transport equation with the parameter \(\tau \) multiplying the second derivatives. First, we show that an explicit two-level vector difference flux-relaxation scheme of the type in [11] for this equation can be reduced to an explicit three-level difference scheme. In this three-level scheme, the parameter \(\tau \) is multiplied by some factor greater than unity in front of the approximation for the second \(t\)-derivative. Second, and most importantly, we analyze the spectral condition for the time-uniform stability of this explicit three-level scheme in the case of constant coefficients. Both sufficient and close-to-them necessary conditions for the validity of the spectral condition are obtained, in particular, in the form of Courant type conditions on the ratio of temporal and spatial steps. These conditions are independent of \( \tau \). The necessary conditions include a condition for the prevalence of the coefficients of viscous terms over the transport coefficients. The derivation is based on analyzing the location of the roots of the corresponding characteristic polynomial with complex coefficients on the complex plane using the generalized Routh–Hurwitz criterion [12, Ch. XV, Sec. 18]. Criteria for a similar spectral stability condition for the differential equation itself and a difference scheme with zero parameter \( \tau \) are preliminarily analyzed as well. Such a criterion for this equation is a condition for a kind of prevalence of the matrix of viscous terms over the vector of transport coefficients; conditions of this kind played an essential role in [4, 10]. For the scheme indicated, the criterion is a Courant type condition. The results are rather encouraging for the subsequent analysis of the corresponding properties of difference schemes for the quasi-gasdynamic system.

1. HYPERBOLIC SECOND-ORDER DIFFERENTIAL EQUATION AND REDUCTION OF AN EXPLICIT TWO-LEVEL VECTOR DIFFERENCE FLUX RELAXATION SCHEME FOR IT TO A THREE-LEVEL SCHEME

Consider the hyperbolic second-order differential equation

$$ \tau \partial _t^2u+\partial _tu+\partial _i\big (b_i(x)u\big )- \tau \,\mathrm {div}\,\big (A(x)\nabla u\big )=f $$
(1)

with a parameter \(\tau >0 \) multiplying the highest derivatives. The sought function \(u=u(x,t) \) is defined and the equation is considered for \(x=(x_1,\ldots ,x_n)\in \mathbb {R}^n\), \(t\geq 0 \), and \(n\geq 1 \). Hereafter, \(A(x)=\mathrm {diag}\,\{a_1(x),\ldots ,a_n(x)\}\) is a diagonal matrix, the operators \(\mathrm {div}\) and \(\nabla \) are taken with respect to \(x \), and summation from 1 to \(n \) over repeated indices \(i \) and \(j \) is assumed.

Let us introduce the uniform mesh \(\omega _{kh} \) with the nodes \(x_{kl}=lh_k \), \(l\in \mathbb {Z}\), and step \(h_k>0 \); the mesh shifted by \(h_k/2 \) with the nodes \(x_{k(l-1/2)}=(l-(1/2))h_k \), \(l\in \mathbb {Z}\); and the mesh operators

$$ s_kv_{l-1/2}=\frac {v_{l-1}\!+v_l}{2},\quad \delta _kv_{l-1/2}=\frac {v_l-v_{l-1}}{h_k},\quad \delta _k^*w_l=\frac {w_{l+1/2}-w_{l-1/2}}{h_k},\quad \Lambda _kv_l\!=\!\frac {v_{l+1}-2v_l+v_{l-1}}{h_k^2}, $$

where \(v_l=v(x_{kl})\) and \( w_{l-1/2}=w(x_{k(l-1/2)})\), \(1\leq k\leq n \). Let us also set \(h=(h_1,\ldots ,h_n) \) and introduce the mesh \(\omega _h:=\omega _{1h}\times \ldots \times \omega _{nh}\) and the operators \( s=(s_1,\ldots ,s_n)\) and \(\nabla _h=(\delta _1,\ldots ,\delta _n)\).

Let us introduce the uniform mesh \(\overline \omega ^{\,h_t} \) with the nodes \(t_m=mh_t \), \(m\geq 0\), and step \(h_t>0 \) and the difference operators

$$ \bar {\delta }_ty=\frac {y-\check {y}}{h_t},\quad \delta _ty=\frac {\widehat {y}-y}{h_t},\quad \mathring {\delta }_ty=\frac {\widehat {y}-\check {y}}{2h_t},\quad \Lambda _ty=\delta _t\bar {\delta }_ty=\frac {\widehat {y}-2y+\check {y}}{h_t^2}, $$

where \(y^m=y(t_m)\), \(\check {y}^{m}=y^{m-1}\), and \(\widehat {y}^{\,m}=y^{m+1} \). Let \(\omega ^{h_t}=\overline \omega ^{\,h_t}\backslash \{0\}\).

For Eq. (1), consider the following explicit vector difference scheme with flux relaxation of the type in [11] that is symmetric and three-point with respect to \(x_1,\ldots ,x_n \) and two-level with respect to \(t \):

$$ \widehat {\varphi }=e^{-h_t/\tau }\varphi -(1-e^{-h_t/\tau })(B sv-\tau A\nabla _hv), $$
(2)
$$ \delta _tv=\mathrm {div}_h\widehat {\varphi }+f. $$
(3)

We seek a function \(v\approx u \) defined on the mesh \(\omega _h\times \overline \omega ^{\,h_t}\) and an auxiliary vector function \(\varphi =(\varphi _1,\ldots ,\varphi _n)^{\mathrm {T}} \) with components \(\varphi _k \) defined on the mesh shifted by \(h_k/2 \) along \(x_k \) with respect to \(\omega _h\times \overline \omega ^{\,h_t}\). Here \(B(x)=\mathrm {diag}\,\{b_1(x),\ldots ,b_n(x)\}\) is a diagonal matrix, \( \mathrm {div}_h\varphi =\delta _1^*\varphi _1+\ldots +\delta _n^*\varphi _n \), and the equations of the scheme are written on the same meshes as the sought functions themselves.

Let us reduce the two-level vector scheme to a three-level scheme of the standard type for \(v \).

Theorem 1.

The function \(v \) satisfies the time-explicit symmetric difference scheme

$$ \tau \alpha _0\biggl (\frac {h_t}{2\tau }\biggr )\Lambda _tv+\mathring {\delta }_{t}v+ \mathrm {div}_h(B sv)-\tau \,\mathrm {div}_h(A\nabla _hv)=\tilde {f}\quad \text {on}\quad \omega _h\times \omega ^{h_t} $$
(4)

that is three-point in all variables \(x_1,\ldots ,x_n \) and \(t \) (three-level in \(t \)), where \(\alpha _0(\zeta ):=\zeta \coth \zeta \) and the free term has the form

$$ \tilde {f}:=\frac {f-e^{-h_t/\tau }\check {f}}{1-e^{-h_t/\tau }}. $$
(5)

Proof. We apply the operator \(\mathrm {div}_h \) to Eq. (2) to obtain the equation

$$ \mathrm {div}_h\widehat {\varphi }=e^{-h_t/\tau }\,\mathrm {div}_h\varphi - (1-e^{-h_t/\tau })\,\mathrm {div}_h(B sv-\tau A\nabla _hv).$$

Let us eliminate the function \(\varphi \) from this equation by expressing \(\mathrm {div}_h\widehat {\varphi }\) and \(\mathrm {div}_h\varphi =\bar {\delta }_tv-\check {f}\) by virtue of Eq. (3) as

$$ \delta _tv-f=e^{-h_t/\tau }(\bar {\delta }_tv-\check {f})- (1-e^{-h_t/\tau })\,\mathrm {div}_h(B sv-\tau A\nabla _hv). $$

We symmetrize the notation in the last equation by multiplying it by \( e^{h_t/(2\tau )}\):

$$ e^{h_t/(2\tau )}\delta _tv-e^{-h_t/(2\tau )}\bar {\delta }_tv+ 2\sinh \frac {h_t}{2\tau }\,\mathrm {div}_h(B sv-\tau A\nabla _hv) =e^{h_t/(2\tau )}f-e^{-h_t/(2\tau )}\check {f}. $$

Further, we use the simple formulas

$$ \delta _tv=\mathring {\delta }_{t}v+(h_t/2)\Lambda _tv,\quad \bar {\delta }_tv=\mathring {\delta }_{t}v-(h_t/2)\Lambda _tv$$

and proceed to the equation

$$ h_t\cosh \frac {h_t}{2\tau }\Lambda _tv+2\sinh \frac {h_t}{2\tau }\mathring {\delta }_{t}v+ 2\sinh \frac {h_t}{2\tau }\,\mathrm {div}_h(B sv-\tau A\nabla _hv) =e^{h_t/(2\tau )}f-e^{-h_t/(2\tau )}\check {f}.$$

We divide the last equation by \( 2\sinh (h_t/(2\tau ))\) and finally obtain the explicit difference scheme (4), (5) three-level in time for \(v \). The proof of the theorem is complete.

Note that \(\alpha _0(\zeta )>\max \{1,\zeta \} \) for \(\zeta >0 \) and \(\alpha _0(\zeta )=\zeta (1+O(e^{-2\zeta }))\) as \(\zeta \to +\infty \); further, \(|\alpha _0(\zeta )-\zeta |<0.075\), \(0.015 \), \(0.0027\) already for \(\zeta \geq 2,3,4\), respectively. Note that, in the limit as \(h_t/(2\tau )\to +\infty \), the expression \(\tau \alpha _0(h_t/(2\tau )) \) is replaced by the \(h_t/2 \), and the three-level difference scheme (4) passes into the explicit two-level scheme

$$ \delta _tv+\mathrm {div}_h(B sv)-\tau \,\mathrm {div}_h(A\nabla _hv)=f, $$

because \((h_t/2)\Lambda _tv+\mathring {\delta }_{t}v=\delta _tv \).

Note that, in the case of constant coefficients, the multiplication of the difference scheme (4) by \(\tau \) and the replacement of the steps \((h,h_t)\mapsto (h/\tau ,h_t/\tau )\) lead to the special case of this scheme with \(\tau =1 \) (including the argument of the coefficient \(\alpha _0(h_t/(2\tau ))\)). Therefore, some properties of this scheme are independent of \(\tau \). This is also true for the spectral condition for time-uniform stability studied below.

It is important that the above method of reducing a two-level scheme to a three-level one can be applied not only to the scalar case in question but also to the quasi-gasdynamic system itself. The reduction of other two-level vector methods for second-order hyperbolic equations to three-level methods was carried out, in particular, in [10, 13, 14]. We also note that numerous two-level difference schemes for the transport equation can be found, e.g., in [15].

2. MULTIDIMENSIONAL TRANSPORT EQUATION WITH PERTURBATIONS AND AN EXPLICIT THREE-LEVEL DIFFERENCE SCHEME FOR IT

Consider the multidimensional homogeneous transport equation perturbed by terms containing the second derivatives of the sought function with respect to \(t \) and \(x_i \) and multiplied by parameters:

$$ \alpha _0\tau \partial _t^2u+\partial _tu+b_i\partial _iu-\tau a_{ij}\partial _i\partial _ju=0, $$
(6)

where \(b_i \) and \(a_{ij} \) are constant coefficients and \(\alpha _0>0 \) and \(\tau >0 \) are parameters. Recall that repeated indices \(i \) and \(j \) assume summation from 1 to \(n \). The introduction of the parameter \(\alpha _0 \) is due to the form of the scheme (4). (We can readily eliminate this parameter by the changes of variables \(\alpha _0\tau \mapsto \tau \) and \(a_{ij}\mapsto a_{ij}/\alpha _0 \); however, we preserve the parameter for clarity.) The sought function \(u(x,t)\) is defined and Eq. (6) is considered on \(\mathbb {R}^n\times [0,+\infty )\). The functions \(u|_{t=0} \) and \(\partial _tu|_{t=0} \) are assumed to be given. Without loss of generality, we assume that the matrix \(A=(a_{ij})_{i,j=1}^n \) is symmetric. Let \(b=(b_1,\ldots ,b_n)^{\mathrm {T}}\).

First, we draw attention to the fact that the differentiation of the transport equation \( \partial _tu+b_i\partial _iu=f\) leads to the equalities

$$ \partial _t^2u+b_j\partial _t\partial _ju=\partial _tf,\quad b_j\partial _j\partial _tu+b_jb_i\partial _j\partial _iu =b_j\partial _jf, $$

and therefore,

$$ \partial _t^2u-b_ib_j\partial _j\partial _iu =\partial _tf-b_j\partial _jf.$$

Thus, the solution of the transport equation also satisfies the second-order equation

$$ \tau \partial _t^2u+\partial _tu+b_i\partial _iu-\tau b_ib_j\partial _i\partial _ju= f+\tau (\partial _tf-b_j\partial _jf) $$

of the type (6) with \(\alpha _0=1\) for any \(\tau \ne 0 \). (For \(\tau =0 \), this is the transport equation itself.) The matrix \(b\otimes b \) with entries \(b_ib_j \) arising in this equation has the property \(b\otimes b\geq 0\), and \(\mathrm {Ker}\,b\otimes b \) coincides with the orthogonal complement to the vector \(b \). Therefore, the resulting equation degenerates for \(n\geq 2 \) as well as for \(n=1 \) and \(b_1=0 \).

Second, the multiplication of Eq. (6) by \(\tau \) and the change of variables \( (x,t)\mapsto (x/\tau ,t/\tau )\) lead to the special case of Eq. (6) with \(\tau =1 \). Therefore, a number of properties of solutions to this equation, including the property analyzed below, are independent of \(\tau \).

Consider complex particular solutions of Eq. (6) of the form

$$ u(x,t)=e^{\mathrm {i}x\cdot \xi }w(\xi ,t),\quad x,\xi =(\xi _1,\ldots ,\xi _n)^{\mathrm {T}}\in \mathbb {R}^n,\quad t\geq 0, $$

where \(\mathrm {i}\) is the imaginary unit and the symbol “\(\cdot \)” stands for the inner product of vectors. Substituting these solutions into Eq. (6), we obtain the ordinary differential equation

$$ \alpha _0\tau \partial _t^2w+\partial _tw+\theta w=0,\quad \theta :=\mathrm {i}b_i\xi _i+\tau a_{ij}\xi _j\xi _i= \mathrm {i}b\cdot \xi +\tau A\xi \cdot \xi \in \mathbb {C}, $$
(7)

for \(w \).

Let us study the uniform boundedness of the solution to this equation with respect to \(t \). Let \(\mathrm {Ker}\,A \) and \(\mathrm {Im}\,A \) be the kernel and image of the matrix \(A \) (i.e., of the corresponding linear operator acting in \(\mathbb {R}^n \)). Recall that since \(A=A^{\mathrm {T}} \), it follows that \(\mathbb {R}^n \) can be represented in the form of the orthogonal direct sum of the subspaces \(\mathrm {Ker}\,A\) and \(\mathrm {Im}\,A\).

Theorem 2.

A necessary condition that the estimate

$$ \sup \limits _{t\geq 0}|w(\xi ,t)|<\infty \quad \text {for each}\quad \xi \in \mathbb {R}^n $$
(8)

be satisfied for any \(w(\xi ,0) \) and \( \partial _tw(\xi ,0)\) is that \(A\geq 0\) .

Along with the condition \( A\geq 0\), the following conditions serve as criteria (i.e., necessary and sufficient conditions) for the estimate (8) to hold:

  1. 1.

    If \(A>0 \), then \( \alpha _0|A^{-1/2}b|^2=\alpha _0A^{-1}b\cdot b\leq 1 \).

  2. 2.

    If, however, \(\mathrm {Ker}\,A\ne \{0\} \), then \( b\in \mathrm {Im}\,A\), and for \(\mathrm {Im}\,A\ne \{0\} \) we have \( \alpha _0\widetilde {A}^{-1}{b}\cdot {b}\leq 1 \), where \( \widetilde {A}\) is the restriction of \(A\) to \(\mathrm {Im}\,A \); otherwise, \(b=0 \) for \( \mathrm {Im}\,A=\{0\}\) (i.e., \(A=0 \)).

In particular, if \( A=\mathrm {diag}\,\{a_1,\ldots ,a_n\} \) , then the criterion is given by the set of conditions

$$ a_k\geq 0,\quad \text {and if }a_k=0\text {, then }b_k=0\text { for }1\leq k\leq n;\quad \alpha _0\sum _{k:a_k>0}\frac {b_k^2}{a_k}\leq 1. $$
(9)

Proof. The characteristic equation for the ordinary differential equation (7) has the form

$$ r_2(z)\equiv \alpha _0\tau z^2+z+\theta =0,\quad z\in \mathbb {C}.$$

Property (8) is equivalent to the property that both roots of the polynomial \(r_2(z)\) lie in the closed left half-plane \( \bar {P}_l:=\{z\in \mathbb {C}{:}\) \(z_R\leq 0\} \) and there exist no multiple roots on the imaginary axis (i.e., \(z=0 \) is not a multiple root, which is exactly the case). Hereafter, we use the standard notation of complex numbers in the form \(z=z_R+\mathrm {i}z_I \).

To analyze the last property, we use the generalized Routh–Hurwitz criterion for polynomials with complex coefficients [12, Ch. XV, Sec. 18]. To this end, we write the auxiliary polynomial \( -\mathrm {i}r_2(\mathrm {i}z)=z+\theta _I+\mathrm {i}(\alpha _0\tau z^2-\theta _R) \) and the fourth-order determinant that is the resultant (up to the sign) of the arising two polynomials with real coefficients,

$$ V_4=\begin {vmatrix} \alpha _0\tau & 0 & -\theta _R & 0 \\ 0&1&\theta _I & 0 \\ 0&\alpha _0\tau & 0 & -\theta _R\\ 0&0&1&\theta _I \end {vmatrix}.$$

Its leading principal minor of the second order \(V_2=\alpha _0\tau \) is positive, and the determinant itself is equal to

$$ V_4=\alpha _0\tau (\theta _R-\alpha _0\tau \theta _I^2) =\alpha _0\tau ^2\big [A\xi \cdot \xi -\alpha _0(b\cdot \xi )^2\big ]. $$

First, let \(V_4\ne 0\). Then the criterion for the roots \(r_2(z)\) to lie in the open left half-plane \(P_l:=\{z\in \mathbb {C}{:} \) \(z_R<0\}\) is the condition \(V_4>0 \). For \(V_4=0 \), as is easy to check, there exists a pure imaginary root \(-\mathrm {i}\theta _I\), and therefore, the second root is \(\mathrm {i}\theta _I-1/(\alpha _0\tau )\in P_l\). Thus, the criterion for both roots to lie in \(\bar {P}_l\) is the condition \( V_4\geq 0\); i.e.,

$$ \alpha _0(b\cdot \xi )^2\leq A\xi \cdot \xi \quad \text {for all}\quad \xi \in \mathbb {R}^n. $$
(10)

It can be written in the form \(\alpha _0 b\otimes b\leq A \), and this is, in a sense, a condition of prevalence of \(A \) over \(b \). This condition does not contain the parameter \(\tau \); this is natural in accordance with the preceding. Conditions of this kind play an essential role in the papers [4, 10].

It follows from the criterion (10) that \(A\geq 0 \). If \(A>0 \), then it can be written in the form

$$ \alpha _0(A^{-1/2}b\cdot A^{1/2}\xi )^2\leq |A^{1/2}\xi |^2\quad \text {for all}\quad \xi \in \mathbb {R}^n.$$

Introducing the vector \(\eta :=A^{1/2}\xi \), we obtain the equivalent explicit criterion in Item 1 of the theorem.

If, however \(\mathrm {Ker}\,A\ne \{0\} \), then \(b\perp \mathrm {Ker}\,A \) by virtue of the criterion (10), hence \(b\in \mathrm {Im}\,A \), and the criterion itself acquires the form

$$ \alpha _0(b\cdot \xi )^2\leq A\xi \cdot \xi =\widetilde {A}\xi \cdot \xi \quad \text {for all}\quad \xi \in \mathrm {Im}\, A;$$

this reduces Item 2 of the theorem to part 1 in which \(\mathrm {Im}\,A\) serves as \(\mathbb {R}^n\) for \(\mathrm {Im}\,A\ne \{0\} \). The case of \(\mathrm {Im}\,A=\{0\} \) is obvious. The proof of the theorem is complete.

Now for Eq. (6) we consider an explicit symmetric difference scheme that is three-point in all variables \(x_1,\ldots ,x_n \) and \(t \) (three-level with respect to \(t \)):

$$ \alpha _0\tau \Lambda _tv+\mathring {\delta }_{t}v+ b_i\mathring {\delta }_iv-\tau a_i\Lambda _iv=0\quad \text {on}\quad \omega _h\times \omega ^{h_t}, $$
(11)

where the sought function \(v \) is defined on \(\omega _h\times \bar {\omega }^{h_t}\). The values \(v(x_k,0) \) and \(v(x_k,h_t) \) at the initial and first levels in time, where \( x_k=(k_1h_1,\ldots ,k_nh_n)\), \(k=(k_1,\ldots ,k_n)\in \mathbb {Z}^n\), are assumed to be given. Here, for clarity, we take the case of \( A=\mathrm {diag}\,\{a_1,\ldots ,a_n\}\) with \(a_i>0 \) for \(1\leq i\leq n \). Since \(\mathrm {div}_h(B s)=b_i\mathring {\delta }_i\) and \(\mathrm {div}_h(A\nabla _h)=a_i\Lambda _i\) for constant coefficients \(b_i \) and \(a_i \), we see that this scheme is obtained from the scheme (4).

Consider complex particular solutions of the difference scheme (11) of the form

$$ v(x_k,t_m)=e^{\mathrm {i}k\cdot \xi }w(\xi ,t_m),\quad k\in \mathbb {Z}^n,\quad m\geq 0,\quad \xi =(\xi _1,\ldots ,\xi _n)^{\mathrm {T}}\in [-\pi ,\pi ]^n.$$

The spectral condition for time-uniform stability is the property

$$ \sup \limits _{m\geq 0}\big |w(\xi ,t_m)\big |<\infty \quad \text {for each}\quad \xi \in [-\pi ,\pi ]^n $$
(12)

for any \(w(\xi ,0) \) and \(w(\xi ,h_t) \) (cf. (8) and [16, Ch. 8, Sec. 25]).

The function \(e^{\mathrm {i}k\cdot \xi } \) is an eigenfunction of the operators \(\mathring {\delta }_l\) and \(\Lambda _l \) with the respective eigenvalues \(\mathrm {i}h_l^{-1}\sin \xi _l \) and \(-4h_l^{-2}\sin ^2(\xi _l/2) \). Therefore, \(w \) satisfies the three-level difference equation (actually, a difference scheme for the ordinary differential equation (7))

$$ \alpha _0\tau \Lambda _tw+\mathring {\delta }_{t}w +\biggl (\mathrm {i}\frac {b_i}{h_i}\sin \xi _i+ \tau \frac {4a_i}{h_i^2}\sin ^2\frac {\xi _i}{2}\biggr )w=0\quad \text {on}\quad \omega ^{h_t}. $$
(13)

By the definitions of the operators \(\Lambda _t \) and \(\mathring {\delta }_{t} \), after the multiplication by \(h_t^2/(\alpha _0\tau ) \), the last equation can be written in the recurrent form

$$ (1+d)\widehat {w}-2(1-\theta )w+(1-d)\check {w}=0, $$
(14)

where

$$ d:=\frac {h_t}{2\alpha _0\tau }>0,\quad \theta \equiv \theta (\xi )= 2\frac {h_t^2a_i}{\alpha _0h_i^2}\sin ^2\frac {\xi _i}{2} +\mathrm {i}\frac {h_t^2b_i}{2\alpha _0\tau h_i}\sin \xi _i\in \mathbb {C}. $$

The characteristic equation for the difference equation (14) has the form

$$ p_2(q)\equiv (1+d)q^2-2(1-\theta )q+1-d=0. $$
(15)

The validity of the spectral stability condition (12) is equivalent to the property that both roots of this equation lie in the closed unit disk \(|q|\leq 1\) in \(\mathbb {C} \) and there exist no multiple roots on the boundary of the disk.

For comparison, let us first study the three-level symmetric scheme

$$ \mathring {\delta }_{t}v+b_i\mathring {\delta }_iv=0\quad \text {on}\quad \omega _h\times \omega ^{h_t}, $$
(16)

which is the special case of the scheme (11) for \(\tau =0 \).

Theorem 3.

For the difference scheme (16), a criterion for the spectral condition of time-uniform stability to be satisfied has the form of the Courant type condition

$$ h_t\sum _{k=1}^n\frac {|b_k|}{h_k}<1. $$
(17)

Proof. For \(\tau =0 \), the scheme (13) implies the recurrent equation

$$ \widehat {w}+2\mathrm {i}h_t\gamma w-\check {w}=0,\quad \gamma \equiv \gamma (\xi )=\frac {b_i}{h_i}\sin \xi _i,$$

which is different from (14). The corresponding characteristic equation \( q^2+2\mathrm {i}h_t\gamma q-1=0 \) has the roots

$$ q_{1,2}= \begin {cases} -\mathrm {i}h_t\gamma \pm \sqrt {1-h_t^2\gamma ^2} &\text {if}\quad h_t|\gamma |\leq 1\\ -\mathrm {i}\left (h_t\gamma \pm \sqrt {h_t^2\gamma ^2-1}\right )&\text {if}\quad h_t|\gamma |>1. \end {cases}$$

In the case of \(h_t|\gamma |>1\), the maximum of \(|q_{1,2}| \) is equal to \(h_t|\gamma |+\sqrt {h_t^2\gamma ^2-1}>1\). In the case of \(h_t|\gamma |\leq 1\), we have \(|q_{1,2}|=1 \), but for \(h_t|\gamma |=1 \) the root becomes multiple. This leads to the condition \( h_t|\gamma |<1\) for all \(\xi \in [-\pi ,\pi ]^n \) and finally (for \(\xi =((\pi /2)\,\mathrm {sgn}\, b_1,\ldots ,(\pi /2)\,\mathrm {sgn}\, b_n) \)) to condition (17). The proof of the theorem is complete.

Now let us proceed to analyzing the difference scheme (11).

Theorem 4.

Let \(a_i>0 \) for \( 1\leq i\leq n\). The validity of the spectral condition for the time-uniform stability of the difference scheme (11) is independent of the parameter \(\tau >0 \).

For the validity of this condition, it is necessary that the following two Courant type conditions on \(h_t \) hold:

$$ \frac {h_t}{\sqrt {\alpha _0}}\left (\,\sum _{k=1}^n\frac {a_k}{h_k^2}\right )^{\!1/2}\leq 1,\quad h_t\sum _{k=1}^n\frac {|b_k|}{h_k}\leq 1; $$
(18)

the second of these conditions almost coincides with condition (17). The following condition is also necessary:

$$ \alpha _0\sum _{k=1}^n\frac {b_k^2}{a_k}\leq 1 $$
(19)

(cf. conditions (9)).

The following condition is sufficient for the indicated spectral condition to hold:

$$ \sum _{k=1}^n\max \bigg \{\frac {h_t^2}{\alpha _0}\frac {a_k}{h_k^2},\alpha _0\frac {b_k^2}{a_k}\bigg \}\leq 1. $$
(20)

For \(n=1 \) , it acquires the form

$$ \frac {h_t}{\sqrt {\alpha _0}}\frac {\sqrt {a_1}}{h_1}\leq 1,\quad \alpha _0b_1^2\leq a_1 $$
(21)

and serves as a criterion.

Remark 1.

Let us give some comments on the conditions in this theorem. The second condition in (18) follows from the first condition and condition (19) by virtue of the Cauchy–Schwarz inequality,

$$ \left (h_t\sum _{k=1}^n\frac {|b_k|}{h_k}\right )^{\!2} \leq \left (\alpha _0\sum _{k=1}^n\frac {b_k^2}{a_k}\right ) \frac {h_t^2}{\alpha _0}\sum _{k=1}^n\frac {a_k}{h_k^2}.$$

As will be proved, the sharper condition

$$ h_t\sum _{k=1}^n\frac {|b_k|}{h_k}\leq \left [\left (2-\frac {h_t^2}{\alpha _0}\sum _{k=1}^n\frac {a_k}{h_k^2}\right ) \frac {h_t^2}{\alpha _0}\sum _{k=1}^n\frac {a_k}{h_k^2}\right ]^{1/2}, $$
(22)

whose right-hand side, by virtue of the first condition in (18), is at most \(1 \), is necessary as well.

Obviously, the inequality

$$ \frac {h_t^2}{\alpha _0}\sum _{k=1}^n\frac {a_k}{h_k^2}+\alpha _0\sum _{k=1}^n\frac {b_k^2}{a_k}\leq 1 $$

serves as a sufficient condition coarser than (20) (cf. the first condition in (18) and condition (19)).

Remark 2.

One can use the 1-norms of the coefficient vectors \(a:=(a_1,\ldots ,a_n) \) and \(b \) and, for \(a\ne 0 \) and \(b\ne 0 \), introduce special mean space steps \(h_a>0 \) and \(h_b>0 \) by the formulas

$$ |a|_1:=\sum _{k=1}^n a_k,\quad |b|_1:=\sum _{k=1}^n|b_k|,\quad h_a:={|a|_1^{1/2}}\left (\,\sum _{k=1}^n\frac {a_k}{h_k^2}\right )^{\!-1/2},\quad h_b:=|b|_1\left (\,\sum _{k=1}^n\frac {|b_k|}{h_k}\right )^{\!-1}.$$

Here \(h_a,h_b\in [h_{\min },h_{\max }]\), where \(h_{\min }=\min \limits _{1\leq k\leq n}h_k\) and \(h_{\max }=\max \limits _{1\leq k\leq n}h_k\). Then the Courant type conditions (18) acquire the form

$$ \sqrt {\frac {|a|_1}{\alpha _0}}\frac {h_t}{h_a}\leq 1,\quad |b|_1\frac {h_t}{h_b}\leq 1. $$

Proof. If the roots of Eq. (15) are multiple, then their modulus is equal to \(\sqrt {|1-d|/(1+d)}\) and is less than 1. If \(\theta =0 \), then the polynomial \(p_2(q) \) has 1 and \({(1-d)}/(1+d) \) as its roots (the latter is obviously less than 1 in modulus). Consequently, in what follows we can assume that \(\theta \ne 0 \), and hence \(p_2(1)=2\theta \ne 0 \), and that these roots are simple.

Let us make the fractional-rational transformation \(q(z)=(z+1)/(z-1) \), which establishes a one-to-one correspondence between the punctured closed unit disk \(\{q\in \mathbb {C}{:} \) \(|q|\leq 1\), \(q\ne 1\} \) and the closed left half-plane \(\bar {P}_l \). The correspondence between the punctured unit circle \(\{q\in \mathbb {C}{:}\) \(|q|=1 \), \(q\ne 1\} \) and the imaginary axis \(z_R=0 \) is also one-to-one; note that the inverse function \( z^{(-1)}(q)=(q+1)/(q-1)\) coincides with the original one. Further, we get

$$ p_2\big (q(z)\big )(z-1)^2=2r_2(z):=2(\theta z^2+2dz+2-\theta ).$$

Here \(r_2(1)=2(1+d)>0 \). By \(z_1 \) and \(z_2 \) we denote the roots of the polynomial \(r_2(z) \) and obtain conditions for them to belong to the half-plane \(\bar {P}_l\).

First, let \(\theta _I=0\). As is well known, for \( \theta _R\ne 0\) one has \(z_1,z_2\in \bar {P}_l \) under the following conditions on the coefficients of the polynomial: \( d/\theta _R\geq 0\) and \((2-\theta _R)/\theta _R\geq 0 \). Since \(d>0 \), this leads to the condition \(0<\theta _R\leq 2 \) (with \(z_1,z_2\in P_l \) for \(\theta _R<2 \) or \(z_1=-d<0 \) and \(z_2=0 \) for \(\theta _R=2 \)).

Now let \(\theta _I\ne 0\) and hence \(\xi \ne 0 \). We again write the auxiliary polynomial

$$ -r_2(\mathrm {i}z)=\theta _Rz^2+\theta _R-2+\mathrm {i}(\theta _Iz^2-2dz+\theta _I) $$
(23)

and the fourth-order determinant that is the resultant (up to the sign) of the two polynomials with real coefficients on the right-hand side,

$$ V_4= \begin {vmatrix} \theta _I & -2d &\theta _I & 0 \\ \theta _R & 0 &\theta _R-2 & 0 \\ 0&\theta _I & -2d &\theta _I \\ 0&\theta _R & 0 &\theta _R-2 \end {vmatrix}.$$

Its leading principal second-order minor \(V_2=2d\theta _R\) is positive for \(\xi \ne 0 \). Let us multiply the second and fourth rows in \(V_4 \) by \(\theta _I/\theta _R \), subtract them from the first and third rows, respectively, and expand the resulting determinant by the first column:

$$ V_4=-\theta _R \begin {vmatrix} -2d & -2{\theta _I}/{\theta _R} & 0 \\ \theta _I & -2d & -2{\theta _I}/{\theta _R} \\ \theta _R & 0 &\theta _R-2 \end {vmatrix} =-4\theta _R\bigg [d^2(\theta _R-2)+\frac {\theta _I^2}{\theta _R}\bigg ] =-4[d^2\theta _R(\theta _R-2)+\theta _I^2].$$

(One can readily see that the final formula also holds for \(\theta _R=0\).) Recall that for \(V_4\ne 0\) the criterion for the roots \(r_2(z) \) to lie in \(P_l \) is given by the condition \(V_4>0 \). For \(V_4=0 \), there exists a pure imaginary root \(z_1=\mathrm {i}y_1\). It is a common root of the two real polynomials on the right-hand side in (23); it readily follows that \(y_1=\theta _I/(d\theta _R) \). Then \(z_2=-\mathrm {i}y_1^{-1}((2/\theta )-1)\), and hence \( z_{2,R}=-d/(\theta _R|\theta |^2)<0\); i.e., \(z_2\in P_l \). Consequently, the criterion for both roots to lie in \(\bar {P}_l \) and for the imaginary axis not to contain multiple roots is given by the condition \(V_4\geq 0\); i.e.,

$$ \theta _R^2+(\theta _I/d)^2\leq 2\theta _R\quad \text {for all}\quad \xi \in [-\pi ,\pi ]^n. $$
(24)

In accordance with the preceding analysis, this condition also holds for \(\theta _I=0 \). Geometrically, the resulting criterion means that \(\theta \) belongs to an ellipse on \(\mathbb {C} \),

$$ (\theta _R-1)^2+(\theta _I/d)^2\leq 1\quad \text {for all}\quad \xi \in [-\pi ,\pi ]^n. $$
(25)

Since \(\theta _I/d=b_i(h_t/h_i)\sin \xi _i\), we conclude that the parameter \( \tau \) does not occur in this criterion. Obviously, the inequalities

$$ 0\leq \theta _R\leq 2,\quad |\theta _I|/d\leq 1\quad \text {for all}\quad \xi \in [-\pi ,\pi ]^n$$

(the condition for \(\theta \) to belong to a rectangle enclosing the ellipse) are a necessary condition for the criterion (25) to hold. The maximum of \( \theta _R\geq 0\) is obviously attained at \(\xi =(\pi ,\ldots ,\pi )\), and the maximum of \(|\theta _I|/d \) is attained at \(\xi =((\pi /2)\,\mathrm {sgn}\, b_1,\ldots ,(\pi /2)\,\mathrm {sgn}\, b_n) \); this leads to conditions (18).

Setting \(\xi =((\pi /2)\,\mathrm {sgn}\, b_1,\ldots ,(\pi /2)\,\mathrm {sgn}\, b_n)\) in (24), we arrive at the inequality

$$ \frac {|\tilde {\theta }_I|}{d}= h_t\sum _{k=1}^n\frac {|b_k|}{h_k}\leq \sqrt {(2-\tilde {\theta }_R)\tilde {\theta }_R}\quad \text {with}\quad \tilde {\theta }_R=\frac {h_t^2}{\alpha _0} \sum _{k=1}^n\frac {a_k}{h_k^2}, $$

i.e., at the necessary condition (22).

Let us also take \(\xi _k=\varepsilon b_kh_k/(a_kh_t) \), \(k={1,\ldots ,n} \), in the criterion (24) and expand both sides in powers of \(\varepsilon \to 0 \); we obtain

$$ O(\varepsilon ^4)+\varepsilon ^2\left (\,\sum _{k=1}^n\frac {b_k^2}{a_k}\right )^{\!2} \leq \frac {\varepsilon ^2}{\alpha _0}\sum _{k=1}^n\frac {b_k^2}{a_k}+O(\varepsilon ^4).$$

Dividing both sides by \(\varepsilon ^2 \) and passing to the limit as \(\varepsilon \to 0 \), we derive the necessary condition (19).

On the other hand, we have \(|\sin \xi _k|=2\sqrt {\sigma _k(1-\sigma _k)} \), where \(\sigma _k:=\sin ^2(\xi _k/2) \) runs through the segment \([0,1] \), and by the Cauchy–Schwarz inequality one has the estimate

$$ \biggl (\frac {\theta _I}{d}\biggr )^{\!2}\leq 2\theta _R\alpha _0\frac {b_i^2}{a_i}(1-\sigma _i).$$

Therefore, canceling by the factor \(2\theta _R\ne 0\) (which is equivalent to \(\sigma :=(\sigma _1,\ldots ,\sigma _n)\ne 0\)), we conclude that the criterion (24) follows from the inequality

$$ \frac {a_i}{\alpha _0}\biggl (\frac {h_t}{h_i}\biggr )^{\!2}\sigma _i+\alpha _0\frac {b_i^2}{a_i}(1-\sigma _i)\leq 1\quad \text {for all}\quad \sigma \in [0,1]^n,\quad \sigma \ne 0. $$
(26)

Its left-hand side is an affine function \(\sigma \) of simple form, and therefore, as is easily seen, this inequality is equivalent to inequality (20), the latter thereby being a sufficient condition for the validity of the criterion.

For \(n=1\), after dividing by \(2\theta _R \), the criterion (24) acquires the form (26), and therefore, conditions (21) become a criterion.

Remark 3.

In addition, note that in the case of coefficients \(a_i\) of arbitrary signs, the situation in Theorem 4 becomes similar to the one indicated in the criterion (9). First, it was shown in the proof of Theorem 4 that the condition \(\theta _R\geq 0\) is necessary in the case \(\theta _I=0 \). For \(\theta _I\ne 0 \) and \(\theta _R<0 \), we have \(V_2<0 \) and \(V_4<0 \); hence one of the roots \(r_2(z) \) lies in the right half-plane \(z_R>0 \). Therefore, in the case \(\theta _I\ne 0 \) it is also necessary that \(\theta _R\geq 0 \) for all \(\xi \in {[-\pi ,\pi ]}^n \). The latter implies the necessity of the condition \(a_i\geq 0 \), \(1\leq i\leq n \). Second, if \(a_k=0 \) for some \(k \), then we take \(\xi _i=0 \) for \(i\ne k \) and \(\sin \xi _k\ne 0 \). Then \(\theta _R=0 \) and \(\theta _I/d=b_k(h_t/h_k)\sin \xi _k\). For \(b_k\ne 0 \), the roots of \(r_2(z)=\mathrm {i}(\theta _Iz^2-2\mathrm {i}dz-\theta _I-2\mathrm {i}) \) are such that \(z_1+z_2=2\mathrm {i}d/\theta _I\); it follows that \(z_{1R}+z_{2R}=0 \), and for \(z_1,z_2\in \bar {P}_l \) we have \(z_{1R}=z_{2R}=0 \). However, the product \(z_1z_2=-1-(2\mathrm {i}/\theta _I)\) should then be real, which is not the case. Hence the condition \(b_k=0\) is necessary in the case \(a_k=0 \).

Therefore, Theorem 4 also remains valid for \(a_i\geq 0 \), \(1\leq i\leq n \), with the only distinction that the sums in conditions (19) and (20) should be taken over all \(1\leq k\leq n \) such that \(a_k>0 \) and one should bear in mind the property that if \(a_k=0 \), then \(b_k=0 \).

In the particular case of \(b=0\), we have \(\theta _I=0 \), and the first condition in (18) in Theorem 4 becomes a criterion. It is similar to the stability condition for the three-level difference scheme with parameter \(\tau \) in [9] in the case of the weight \(\sigma =0 \). In this case, one should take \(B_h=\alpha _0 I \), \(B_{1h}=I \) (here \(I \) is the identity operator), and \(A_h=-\tau a_i\Lambda _i \) in this scheme, with the stability condition having the form \( h_t(\alpha _h/2)\leq \varepsilon _1\) with an arbitrary \( 0<\varepsilon _1<1\), where

$$ \alpha _h:=\frac {1}{\sqrt {\alpha _0}} \left (\,\sum _{k=1}^n\frac {4a_k}{h_k^2}\sin ^2\frac {\zeta _k}{2}\right )^{\!1/2} <\frac {2}{\sqrt {\alpha _0}}\left (\,\sum _{k=1}^n\frac {a_k}{h_k^2}\right )^{\!1/2},\quad \zeta _k:=\frac {\pi (N_k-1)}{N_k}, $$

\(N_k\) being the number of partitioning segments along \(x_k\) (unlike the present paper, the paper [9] considers an initial–boundary value problem), with \(\sin ^2(\zeta _k/2)\to 1 \) and the last inequality becoming the equality as \(N_k\to +\infty \), \(1\leq k\leq n \). The case of \(\varepsilon _1=1 \) is also possible (with some relaxation of the norms used); in this connection, see [17, Remark 1].