Abstract
In this paper, a compact difference scheme is established for the heat equations with multi-point boundary value conditions. The truncation error of the difference scheme is \(O(\tau ^2+h^4),\) where \(\tau\) and h are the temporal step size and the spatial step size. A prior estimate of the difference solution in a weighted norm is obtained. The unique solvability, stability and convergence of the difference scheme are proved by the energy method. The theoretical statements for the solution of the difference scheme are supported by numerical examples.
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1 Introduction
The classical type of conditions is referred to as local conditions when the values of the unknown function or its derivative are specified only at the boundary points of the problem domain, while the non-local boundary conditions are proposed where the values of the unknown function at all or some points inside the problem domain take part in the condition formulation. The development of numerical methods for the solution of non-local boundary value problems has been a very important research area. There are many models and works about non-local boundary conditions, such as elliptic equations [10, 15], elliptic–parabolic equations [3, 4], hyperbolic equations [2, 7, 16, 17], hyperbolic–parabolic equations [5] and parabolic equations [1, 8, 9, 11,12,13].
We recall two cases of non-local boundary conditions. The first one is the integral boundary conditions
where \(\alpha ,\beta ,g_1,g_2\) are known functions. Another one is the multi-point boundary value conditions
where \(\alpha _r,\beta _s,g_1,g_2\) are known functions.
There is a lot of research on the integral boundary conditions and multi-point boundary conditions under different models. Sun considered the heat equations with integral boundary conditions (1) and got unconditional solvability and \(L_\infty\) convergence for the difference scheme which was second order in time and fourth order in space [13]. Martin-Vaquero and Vigo-Aguiar provided a compact difference scheme for the same problem by the fourth-order Simpson’s composite formula and Crandall’s formula [11]. They improved the accuracy of this algorithm and studied the convergence later in [12]. They all used the composite Simpson rule to approximate the boundary conditions. Yildirim and Uzun established stable difference schemes with third and fourth order for the hyperbolic multi-point non-local boundary value problem [16]. They provided stability estimates and numerical analysis for the solutions of the difference schemes. Ashyralyev and Gercek considered a finite difference method for solving the multi-point elliptic–parabolic partial differential equation and obtained stability, and coercive stability for the solution of the difference scheme [4]. Alikhanov studied multi-point boundary conditions (2) for the heat equation with variable coefficients in the differential and finite-difference settings [1]. He established the difference scheme which is second order both in space and in time. Using the method of energy inequalities, prior estimates for the corresponding differential and finite-difference problems are obtained. Due to the characteristic of the multi-point boundary condition, he just proved the prior estimates in a weighted \(L^2\) norm.
Our work is a good supplement to the previous researches. In this article, we construct a compact difference scheme for the multi-point boundary value problem of the heat equation taking the form of
where \(a>0\) and \(b>0\) are given constants, \(\alpha _r(t),\)\(\beta _s(t),\)\(\mu _1(t),\)\(\mu _2(t)\in C[0,T],\)\(0<\xi _1<\xi _2<\cdots<\xi _M<L,\)\(0<\eta _1<\eta _2<\cdots<\eta _N<L,\)f and \(\varphi\) are continuous functions. We establish a compact difference scheme with the truncation error \(O(\tau ^2+h^4)\) and get a weighted L2 norm prior estimate. Then, we prove the unique solvability, convergence and stability using the energy method.
The rest of this paper is organized as follows: some notations are introduced and several important lemmas are given in Sect. 2. Then, a compact difference scheme is constructed in Sect. 3 and a prior estimate is provided in Sect. 4. Based on a prior estimate, the unique solvability, stability and convergence are proved in Sect. 5. Besides, a compact finite-difference scheme is also given for the multi-point boundary value problems of the heat equation with variable coefficients in Sect. 6. At last, two numerical examples are presented in Sect. 7 and a brief conclusion is given in Sect. 8, respectively.
2 Preliminary
In this section, some useful notations and lemmas will be prepared.
For finite-difference approximation, we discretize equally the interval [0, L] with \(x_i=ih\;(0\le {i}\le {m}),\) [0, T] with \(t_k=k\tau \;(0\le {k}\le {n}),\) where \(h=L/m\) and \(\tau =T/n\) are the spatial and temporal step sizes, respectively. Denote \(t_{k+\frac{1}{2}}=(t_k+t_{k+1})/2,\)\(\Omega _h=\{x_i\;|\;0\le {i}\le {m}\},\)\(\Omega _\tau =\{t_k\;|\;0\le {k}\le {n}\},\) then the computational domain \([0,L]\times [0,T]\) is covered by \(\Omega _h\times \Omega _\tau .\) For any mesh function \(v=\{v_i^k\;|\;0\le {i}\le {m},\;0\le {k}\le {n}\}\) defined on \(\Omega _h\times \Omega _\tau ,\) introduce the following notations:
Let \(v^k=(v_0^k,v_1^k,\ldots ,v_m^k),\) then \(v^k\) is a mesh function defined on \(\Omega _h.\)
Denote
and
For any \(v\in V_h,\) introduce the following norms or seminorms:
For any \(0<c<d<L,\) define
With an assumption that \(2h<d-c,\) we give similar definitions in the interval [c, d]:
For any grid function \(w\in V_h,\) define
We need some lemmas for establishing and analyzing the difference scheme for (3)–(5).
Lemma 2.1
[14] Let\(v\in V_h.\)Then, for any\(\varepsilon >0,\)we have
Lemma 2.2
Let \(v\in V_h\) and \(0<c<d<L.\) If \(h<(d-c)/4,\) then for any \(\varepsilon >0,\) we have
where \(c_0=\min \{c(L-c), d(L-d)\}.\)
Proof
According to Lemma 2.1, we have
This completes the proof.
Lemma 2.3
For any grid function \(v\in V_h,\) we have
Proof
From the definition, we can get
In obtaining the last inequality, we have used \(2p_i^2-p_{i-1}^2-p_{i+1}^2>0\) when \(1\le i\le m-1.\)
This completes the proof.
Lemma 2.4
Suppose\(f(x)\in C[d,d+3h].\)Taking\(d,\;d+h,\;d+2h,\;d+3h\)as the interpolation points, we obtain the third-order interpolation polynomial off(x):
Then, it satisfies
Proof
Let
We have
Therefore, we obtain
This completes the proof.
Lemma 2.5
[14] Let\(h>0\)andcbe two constants. Suppose\(g(x)\in {C}^6[c-h,c+h].\)Then,
Lemma 2.6
[6] Let\(\{F^k\;|\;k\ge 0\}\)and\(\{G^k\;|\;k\ge 0\}\)be two nonnegative sequences and satisfy
wherecis a nonnegative constant. Then, we have
3 Derivation of the Difference Scheme
Define a grid function
on \(\Omega _h\times \Omega _\tau ,\) where
Suppose h is small enough satisfying \(\xi _1>2h,\)\(\xi _M<L-2h,\)\(\eta _1>2h,\) and \(\eta _N<L-2h,\) that is, \(h <\frac{1}{2} \min \{\xi _1, L-\xi _M, \eta _1, L-\eta _N\}.\)
Considering the differential equation (3) at point \((x_i,t_{k+\frac{1}{2}}),\) we have
By the Taylor expansion, we get
where \(f_i^{k+\frac{1}{2}}=f(x_i,t_{k+\frac{1}{2}}).\) Acting the operator \({\mathcal {A}}\) on the above equation, we obtain
Using Lemma 2.5, we have
Substituting (8) into (7), we obtain
There exists a constant \({\hat{c}}_1\) such that
Considering boundary value conditions (4) at \(t_{k+\frac{1}{2}},\) we have
For any \(\xi _r,\) there exists a unique \(i_r\) such that \(\xi _r\in [x_{i_r+1}, x_{i_r+2}).\) Taking \(x_{i_r},\)\(x_{i_r+1},\)\(x_{i_r+2},\)\(x_{i_r+3}\) as the interpolation points, we obtain the third-order interpolation polynomial of u(x, t):
and we have
Similarly, for any \(\eta _s,\) there is a unique \(j_s\) satisfying \(\eta _s\in [ x_{j_s+1}, x_{j_s+2}).\) Taking \(x_{j_s},\)\(x_{j_s+1},\)\(x_{j_s+2},\)\(x_{j_s+3}\) as interpolation points, we obtain the third-order interpolation polynomial of u(x, t):
and we have
Substituting (13) into (11) and (14) into (12), we get
or
and there exists a constant \({\hat{c}}_2\) such that
Noticing the initial condition (5),
omitting the small items \(R_i^{k+\frac{1}{2}}\;(0\le {i}\le {m})\) in the formula (9), (15) and (16), and replacing \(U_i^k\) by \(u_i^k,\) we obtain the following difference scheme:
4 Prior Estimates
Lemma 4.1
The solution of (18)–(21) satisfies the following equality:
Proof
Multiplying equality (18) by \(h{\mathcal {A}}u_i^{k+\frac{1}{2}}\) and summing the result with respect to i from \(\eta\) to \(\xi ,\) we obtain
We multiply the above equality by h and sum up for \(\xi\) from \(\eta\) to \(m-1,\) and then multiply the result by h and sum up for \(\eta\) from 1 to \(m-1\) to get
Due to
and
we obtain
Some calculation yields
Substituting the above equality into (25), we can get (22).
This completes the proof.
Theorem 4.1
Let\(\{u_i^k\;|\;0\le {i}\le {m},0\le {k}\le {n}\}\)be the solution of (18)–(21). Denote
Whenhand\(\tau\)are small enough, we have
Proof
By Lemma 4.1 and inequalities
we have
According to Lemmas 2.2 and 2.3, when \(4h\le \min \{\xi _1,L-\xi _M,\eta _1,L-\eta _n,\xi _M-\xi _1,\eta _N-\eta _1\},\) we have
Taking \(\varepsilon =1/(6L(\alpha _0+\beta _0)),\) we have
When \(h<\frac{c_0}{2}\,\sqrt{\frac{3c_1}{(\alpha _0+\beta _0)L[3c_1(\alpha _0+\beta _0)L+c_0]\,}}\,,\) we get
Substituting the above inequality into (28), we obtain
That is
When \(c_2\tau \le 1/3,\) we have
This completes the proof.
5 The Unique Solvability, Stability and Convergence
5.1 Unique Solvability
Theorem 5.1
Difference scheme (18)–(21) has a unique solution.
Proof
The difference scheme (18)–(21) is a linear system of algebraic equations. Let \(u^k=(u_0^k,u_1^k,\ldots ,u_m^k).\) According to (21), we obtain the value of \(u^0.\) If the value \(u^k\) of the k-th time level is obtained, then we can obtain the value of \(u^{k+1}\) through (18)–(20). Consider the homogeneous system about \(u^{k+1}\):
According to Lemma 4.1 and similar to the derivation of (29), we get the result
When \(\tau <1/c_2,\) we have \(\Vert p({\mathcal {A}}u^{k+1})\Vert _0^2=0,\) that implies that
Then, it follows from (30) that
Both above equalities yield that
Combining with (31) and (32), we know \(u_i^{k+1}=0,\,0\le i\le m.\) Thus, the homogeneous system only has a trivial solution.
This completes the proof.
5.2 Stability and Convergence
According to Theorem 4.1, we can obtain the following result easily.
Theorem 5.2
The difference scheme (18)–(21) is stable to the initial value and the right term in the sense that: let\(\{u_i^k\;|\;0\le {i}\le {m},0\le {k}\le {n}\}\)be the solution of difference scheme (18)–(21), then we have
where\(c_2\)and\(c_3\)are defined in (26).
Theorem 5.3
The finite-difference scheme (18)–(21) is convergent with the convergence order of\(O(\tau ^2+h^4)\)in the weighted norm.
Proof
Let
Then, subtracting (18)–(21) from (9), (15), (16) and (17) yields the error equations
From Theorem 4.1, we obtain
This completes the proof of the theorem.
6 A Compact Difference Scheme for Heat Equations with Variable Coefficients
We have discussed a compact difference scheme for the heat equation with the constant coefficients in above sections. In this section, we will consider the heat equation with variable coefficients in [1]:
Let
Then, the function u(x, t) satisfies that
where \(c(x,t)=k''(x)+(k'(x))^2+b(x)k'(x)/a(x)+{\bar{c}}(x,t)/a(x),\;f(x,t)=e^{-k(x)}g(x,t).\) The convection term disappears now. Thus, we consider the multi-point boundary value problem of the heat equation with variable coefficients:
Similar to the establishment of (18)–(21) for problem (3)–(5), we present a compact difference scheme for (37)–(39) as follows:
The truncation errors of (40)–(42) are all \(O(\tau ^2+h^4).\) The proof of solvability, stability and convergence of (40)–(43) is similar to that of (18)–(21), so we do not repeat it here and just show a numerical example in Sect. 7.
7 Numerical Tests
Example 1
Use the compact difference scheme (18)–(21) to solve the following problem
The exact solution is \(u(x,t)=e^{x-t}.\) Define
Table 1 presents the errors in a weighted \(L^2\) norm when we take different space step sizes. When the time step size is fixed to 1/16 000 and the space step size shrinks, the spatial convergence order is 4. Table 2 lists the errors in a weighted \(L^2\) norm when we take different time step sizes. When the space step is fixed to 1/1 600 and the time step shrinks, the temporal convergence order is 2. Table 3 provides the errors in a weighted \(L^2\) norm when we take different space and time step sizes. The numerical results are consistent with the theoretical analysis of convergence and stability.
Example 2
Use the compact difference scheme (40)–(43) to solve the following problem:
The exact solution is \(u(x,t)=e^{x-t}.\)
Table 4 presents the errors in a weighted \(L^2\) norm when we take different space step sizes with fixing time step size, which shows the spatial convergence order is 4. Table 5 lists the errors in a weighted \(L^2\) norm when we take different time step sizes with fixing space step, which presents the temporal convergence order is 2. Table 6 provides the errors in a weighted \(L^2\) norm when we take different space and time step sizes. The numerical results are consistent with the truncation errors.
8 Conclusion
In this article, a compact difference scheme is constructed to solve the multi-point boundary value problem of the heat conduction equation with constant coefficients. Using the energy method, the prior estimate is obtained and the unique solvability, stability and convergence are proved rigorously. Because of the complexity of the multi-point boundary conditions, the convergence order \(O(\tau ^2+h^4)\) is obtained only in a weighted \(L^2\) norm. Besides, a compact difference scheme is also constructed for the problem with variable coefficients. Numerical examples are provided to confirm the accuracy of the difference scheme, which are consistent with the theoretical analysis. In the future, efforts will be taken to perform an analysis on the difference scheme in \(L^2\) norm and in \(L_\infty\) norm.
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Acknowledgements
The research is supported by the National Natural Science Foundation of China (No. 11671081) and the Fundamental Research Funds for the Central Universities (No. 242017K41044).
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Wang, X., Sun, Z. A Compact Difference Scheme for Multi-point Boundary Value Problems of Heat Equations. Commun. Appl. Math. Comput. 1, 545–563 (2019). https://doi.org/10.1007/s42967-019-00025-w
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DOI: https://doi.org/10.1007/s42967-019-00025-w