Abstract Generated abstract
The paper studies convergence of a finite difference scheme for the multidimensional heat conduction equation with variable coefficients under mixed initial and boundary conditions. Using the equivalence theorem for stability and consistency, the authors analyze the step operator through a sequence of matrix lemmas concerning symmetric operator pencils, eigenvalues, and comparison with constant coefficient bounds. They prove that stability of the variable coefficient scheme can be inferred from associated constant coefficient operators, and derive a sufficient convergence criterion: unconditional convergence in the mean for one half less than or equal to the weight parameter and at most one, and a mesh dependent restriction for smaller weights. The approach is also noted to extend to certain nonlinear and hyperbolic problems.
Full Text
MATHEMATICS
N. N. YANENKO and Yu. E. BOYARINTSEV
ON THE CONVERGENCE OF DIFFERENCE SCHEMES FOR THE HEAT-CONDUCTION EQUATION WITH VARIABLE COEFFICIENTS
(Presented by Academician S. L. Sobolev on 31 III 1961)
1. For the equation
\[ \frac{\partial u}{\partial t} = \sum_{i=1}^{m} \frac{\partial}{\partial x_i} \left[ \chi(x_1,x_2,\ldots,x_m,t)\frac{\partial u}{\partial x_i} \right], \quad \chi(x_1,x_2,\ldots,x_m,t)\ge \chi_0>0; \tag{1} \]
we pose the mixed Cauchy problem
\[ u(x_1,x_2,\ldots,x_m,0)=\varphi(x_1,x_2,\ldots,x_m); \tag{2} \]
\[ \begin{aligned} u(x_1,x_2,\ldots,x_{s-1},0,x_{s+1},\ldots,x_m,t) &=f_s(x_1,\ldots,x_{s-1},x_{s+1},\ldots,x_m,t),\\ u(x_1,x_2,\ldots,x_{s-1},1,x_{s+1},\ldots,x_m,t) &=g_s(x_1,\ldots,x_{s-1},x_{s+1},\ldots,x_m,t). \end{aligned} \tag{3} \]
We shall require sufficient smoothness of the functions \(\chi(x,t)\), \(\varphi(x)\), \(f_s(x,t)\), \(g_s(x,t)\).
For the solution of problem (1)—(3) we construct the following difference scheme:
\[ A_2(ar^{n+1})u^{n+1}=A_1[(1-\alpha)r^n]u^n, \qquad 0\le \alpha\le 1; \tag{4} \]
\[ u^0_{i_1\ldots i_m}=\varphi_{i_1\ldots i_m} = \varphi(i_1h_1,\ldots,i_mh_m); \tag{5} \]
\[ u^n_{i_1\ldots i_{s-1},0,i_{s+1}\ldots i_m} = f^n_{s i_1\ldots i_{s-1},i_{s+1}\ldots i_m}, \tag{6} \]
\[ u^n_{i_1\ldots i_{s-1},N_s+1,i_{s+1}\ldots i_m} = g^n_{s i_1\ldots i_{s-1},i_{s+1}\ldots i_m}, \]
where it is put that
\[ u^n=\{u^n_{i_1\ldots i_m}\} = u(i_1h_1,\ldots,i_mh_m,n\tau); \]
\[ i_s=0,1,\ldots,(N_s+1); \qquad (N_s+1)h_s=1; \]
\[ r^n=\{r^n_{s i_1\ldots i_m}\}; \qquad r^n_{s i_1\ldots i_m} = \frac{\chi^n_{i_1\ldots i_s+1/2\ldots i_m}\tau}{h_s^2}; \]
\[ A_2(ar)=E-\alpha B(r); \qquad A_1[(1-\alpha)r]=E+(1-\alpha)B(r); \]
\[ B(r)=\sum_{s=1}^{m}B_s; \qquad B_s=\{B^j_{si}\}; \tag{7} \]
\[ B^j_{si} = r_{s i_1\ldots i_s-1\ldots i_m} \delta^{j_1\ldots j_s\ldots j_m}_{i_1\ldots i_s-1\ldots i_m} - \]
\[ - \left( r_{s i_1\ldots i_s-1\ldots i_m} + r_{s i_1\ldots i_s\ldots i_m} \right) \delta^{j_1\ldots j_s\ldots j_m}_{i_1\ldots i_s\ldots i_m} + r_{s i_1\ldots i_s\ldots i_m} \delta^{j_1\ldots j_s\ldots j_m}_{i_1\ldots i_s+1\ldots i_m}, \]
\[ \delta^{j_1\ldots j_m}_{i_1\ldots i_m} = \delta^{j_1}_{i_1}\cdots \delta^{j_m}_{i_m}, \quad \delta^j_i \text{ is the Kronecker symbol.} \]
By the equivalence theorem \((^{1})\), for convergence of the difference scheme (4)—(6) it is necessary and sufficient that the step operator of equation (4)
\[ C_n=A_2^{-1}(ar^{n+1})A_1[(1-\alpha)r^n] \tag{8} \]
have a norm satisfying the condition
\[ \|C_n\|=1+d_n\tau,\qquad |d_n|\leq K; \tag{9} \]
the constant \(K\) does not depend on \(n,\tau,h\). We shall show that in our case this is so.
§ 2. We first formulate a number of lemmas.
Lemma 1. Let \(A,B\) be two symmetric matrices; let \(C(\varepsilon)\) be the matrix of the pencil
\[ C(\varepsilon)=A+\varepsilon B; \tag{10} \]
\(x_i(\varepsilon)\) the \(i\)-th normalized eigenvector of \(C(\varepsilon)\); \(\lambda_i(\varepsilon)\) the corresponding eigenvalue.
Then \(x_i(\varepsilon)\), \(\lambda_i(\varepsilon)\) satisfy the following system of equations
\[ \frac{d\lambda_i}{d\varepsilon}=(Bx_i,x_i); \tag{11} \]
\[ \frac{dx_i}{d\varepsilon} = \sum_{k=1,k\ne i}^{n} \frac{(Bx_i,x_k)}{\lambda_i-\lambda_k}\,x_k. \tag{12} \]
Lemma 2. If
\[ r_{s i_1\ldots i_m}\geq r_0>0,\qquad 0\leq \alpha\leq 1, \tag{13} \]
then the matrix \(B(r)\) has nonnegative characteristic roots, the matrix \(A_2(\alpha r)\) has positive roots not less than 1, and the matrix \(A_1[(1-\alpha)r]\) has roots not exceeding 1.
Lemma 2 follows from Lemma 1.
Lemma 3. Let
\[ r_s=\min_{i_1\ldots i_m}\{r_{s i_1\ldots i_m}\},\qquad R_s=\max_{i_1\ldots i_m}\{r_{s i_1\ldots i_m}\};\qquad 0\leq \alpha\leq 1, \]
\[ b_s=B(r_s);\qquad B_s=B(R_s),\qquad b=\sum_{s=1}^{m} b_s,\qquad B=\sum_{s=1}^{m} B_s; \tag{14} \]
\[ a_2=E-\alpha b;\qquad a_1=E+(1-\alpha)b;\qquad c=a_2^{-1}a_1; \]
\[ A_2=E-\alpha B,\qquad A_1=E+(1-\alpha)B,\qquad C=A_2^{-1}A_1. \]
Then for the matrix
\[ C(r)=A_2^{-1}(\alpha r)A_1[(1-\alpha)r] \tag{15} \]
the relation
\[ \|c\|\leq \|C(r)\|\leq \|C\| \tag{16} \]
is valid.
Lemma 4. Put
\[ \widetilde C_n=A_2^{-1}(\alpha r^n)A_1[(1-\alpha)r^n]. \tag{17} \]
Then for \(C_n\) (8) one has
\[ \|C_n\|=\|\widetilde C_n\|(1+l_n\tau),\qquad |l_n|\leq K, \tag{18} \]
where \(K\) does not depend on \(n,\tau,h\).
On the basis of Lemmas 1–4 one may formulate the following theorem.
Theorem 1. The step operator \(C_n\) of scheme (4) is stable if the operator \(C\) is stable, and unstable if the operator \(c\) is unstable.
Since \(c, C\) are operators with constant coefficients, the Neumann criterion is applicable for determining their stability. Therefore the following sufficient convergence criterion can be formulated:
Theorem 2. If \(1/2 \leqslant \alpha \leqslant 1\), or \(0 \leqslant \alpha < 1/2\) and
\[ \sum_{s=1}^{m} R_s \leqslant \frac{1}{2-4\alpha}, \tag{19} \]
then the solution of (4)—(6) converges to the solution of (1)—(3) in the mean.
Let us note that Lemma 3 and Theorems 1 and 2 are also valid for nonlinear equations. However, in this case the \(R_s\) depend on the initial data and the mesh and cannot be estimated effectively.
The indicated method is also applicable to hyperbolic equations.
Received
2 I 1961
References
- R. D. Richtmyer, Difference Methods for Initial-Value Problems, 1960, p. 57.