Uniformly Convergent Scheme for Singularly Perturbed Space Delay Parabolic Differential Equation with Discontinuous Convection Coefficient and Source Term
A singularly perturbed delay parabolic problem of convection-diffusion type with a discontinuous convection coefficient and source term is examined. In the problem, strong interior layers and weak boundary layers are exhibited due to a large delay in the spatial variable and discontinuity of convection coefficient and source. The problem is discretized by a nonstandard finite difference scheme in the spatial variable and for the time derivative, we used the Crank–Nicolson scheme. To enhance the order of convergence of the spatial variable, the Richardson extrapolation technique is applied. The error analysis of the proposed scheme was carried out and proved that the scheme is uniformly convergent of second order in both spatial and temporal variables. Numerical experiments are performed to verify the theoretical estimates.
In many real life situations, we encounter problems with having small parameters multiplying the highest order derivative terms, involving at least one shift term. We call these singularly perturbed delay differential equations. Such problems arise frequently in the mathematical modeling of various physical and biological phenomena. The delay terms in the models enable us to include some past behavior to get more practical models for the phenomena. For example, Stein’s model is a well-known space dependent model which represents a commonly-used description of spontaneous neuronal activity . Many other examples can be found in [2, 3]. Extensive numerical methods have been developed for singularly perturbed delay differential equations, such as [4–7] and references therein.
Nonetheless, there are numerical methods for singularly perturbed ordinary differential equations with nonsmooth data (discontinuous source term and/or convection coefficient) using special piecewise uniform meshes; see [12–16] and references therein.
When we came to time dependent differential equations, in  the authors studied singularly perturbed parabolic delay differential equation with discontinuous coefficient and source term based on the upwind finite difference method on a specially generated mesh in the spatial direction with backward Euler method for the discretization of the time variable. Providing uniform numerical method for singularly perturbed differential equations with discontinuous coefficients and source terms is not that easy. In this case, the situation is more complicated, especially when the delay is large.
Motivated by the above-given works, we have designed a numerical scheme for singularly perturbed time dependent delay differential equation with discontinuous convection coefficient and source therm, using a nonstandard finite difference scheme in the spatial variable and Crank–Nicolson method in a temporal variable. To increase the order of convergence of the spatial variable, the Richardson extrapolation technique is applied. It is proved that the proposed scheme is uniformly convergent of order .
The paper is organized as follows: in Section 2, we formulate the problem. In Section 3, the bounds on the solution and its derivatives is discussed. Section 4 shows the derivation of the numerical scheme. In Section 5, the convergence of the full discrete scheme is analyzed. In Section 6, we describe the Richardson extrapolation technique and its convergence analysis. Numerical results are presented in Section 7, and lastly, conclusions are given in Section 8.
2. Problem Formulation
Consider the following singularly perturbed delay parabolic problems with a discontinuous convection coefficient and source term:subject to the following initial condition and interval boundary conditionswhere is the perturbation parameter, , the function and are sufficiently smooth functions such that and for all . Moreover, assume thatwhere and . The solution of (1) satisfies and at , here and are left and right side limit of at .
The problem defined in (1) can be rewritten as follows:wherewith initial and boundary condition
3. Bounds on the Solution and Its Derivatives
In this section, the analytical aspects of the solution of problem (1) and its derivatives are studied. The differential operator satisfies the following minimum principle.
Lemma 1 (Minimum principle). Suppose and assume that , and and . Then, .
Proof. Define a test functionNote that, and . LetThen, there exists such that and , . Therefore, the function attains minimum at . Suppose the theorem does not hold true, then . Case 1: Case 2: Case 3: In all the cases we reached at a contradiction. Hence, the required result follows.
Lemma 2. (Stability result). If satisfies problem (1), then the bound
Proof. Defining the following barrier functionsand using the minimum principle in Lemma 1 we can obtain the required estimate.
4. Description of the Numerical Scheme
To obtain the totally discrete scheme, we discretized the temporal variable and space variable separately, then formulate the fully discrete scheme.
4.1. Temporal Semidiscretization
On the time domain , we use uniform mesh given by: , where M is the number of mesh intervals in and is the time step. Then, on the continuous problem (1), is discretized by using the Crank–Nicolson method, defined by the following equation:where
After some rearrangement, the temporal discretized form of (1) is given by the following equation:wherewhereand is the semi-discrete approximation to the exact solution of (1) at the time level. The local truncation error of the semi-discrete method (16) is given by the following equation:where is the solution evaluated after one step of the semi-discrete scheme (16) taking the exact value instead of as the initial data.
Lemma 3. Let be a smooth function such that , , and . Then, .
Proof. Apply the same procedure as Lemma 1.
Lemma 4. If satisfies problem (16), then the bound
Proof. Consider a barrier functionClearly, and . Also, for we have two cases. Case 1: Case 2: Therefore, from Lemma 3 we get for all .
Lemma 5. Suppose that , the local truncation error associated to scheme (16) satisfies:
Proof. Using Taylor’s series expansion, expand and centered at , we get the following equation:Then, substitute (25) into (1), we obtain the following equation:whereFrom (26), the local truncation error is the solution of the following BVP:Next, using the maximum principle for the operator proves the result, for further detail one can refer to .
Theorem 1. (Global error estimate). Under the hypothesis of Lemma 5, the global error estimate , associated with the Crank–Nicolson scheme in the time direction at time level is given by the following equation:
Proof. Using the local error estimate given by Lemma 5, we obtain the following global error estimates at time level:
Lemma 6. The solution of the semidiscretized problem (16) satisfies the following equation:for.
4.2. Spatial Discretization
On the spatial domain [0, 2], is discretized uniformly as follows:where is the number of mesh intervals in the spatial variable. Thus, the discretized domain is defined as . In the spatial discretization, the problem (16) is further discretized using the nonstandard finite difference method. The main idea of the nonstandard finite difference method is to replace the denominator of the finite difference approximation of the derivatives by positive functions.
4.3. Nonstandard Finite Difference
For the construction of an exact finite difference scheme, consider the following constant coefficient homogeneous differential equation, corresponding to (16):where . Now, the homogeneous differential equation (33) possesses two linearly independent solutions, given as follows:
The target is to find a difference equation which has the same general solution as the differential equations in (16) at the mesh point xi given by . Using the theory of difference equations in , by taking there consecutive points we can get the following set of equations:or,
Simplifying and substituting the values of and , we obtain the exact difference scheme(in the sense that it has the same general solution as the corresponding differential equation) for (33) as follows:
The main goal is finding a suitable denominator function for the second order derivative, the extraction of the denominator function from equation (38) is not straightforward. As a result, assume layer behaviors of the solution of problem (1) and that of the problem in the case when are similar, so that in equation (38) use the approximation and following steps in , we obtain the following equation:
Therefore, the denominator function for the second order derivative approximation becomes
This denominator function can be modified to variable coefficient problem as follows:
Remark 1. For the discretization of the first derivative of the spatial variable we use backward and forward finite differences depending on the convection coefficient term,
4.4. Fully Discrete Scheme
5. Uniform Convergence of the Fully Discrete Scheme
Lemma 7 (Discrete Minimum principle). Let be a mesh function such that for all and . Then, for all .
Proof. Define a test functionwhere . Note that, and . LetThen, there exists such that and , . Therefore, the function attains a minimum at . Suppose the theorem does not hold true, then . Case 1: Case 2: Case 3: in all cases, the result contradicts with our assumption. Hence, the required result follows.
An immediate consequence of the discrete minimum principle is the following uniform stability property of the operator .
Lemma 8. The solution of the discrete scheme in (25) satisfies the bound
Proof. Define barrier functionthen following the procedure applied for the proof of Lemma 7, we obtain and . The required result follows from Lemma 7.
5.1. Error Estimates
Theorem 2. Let and be sufficiently smooth functions so that . Then, the truncation error of the discrete scheme satisfies the bound.
Proof. The truncation error in the spatial discretization isdepending on the behavior of , the estimate . Then apply Lemma 6 in to the truncation error bound in (53). Case 1: When Case 2: When , applying the same procedure like case 1, we will get the following equation:
Lemma 9. For a fixed mesh, as , it holdswhere .
Proof. For the proof see .