A Modified Algorithm Based on Haar Wavelets for the Numerical Simulation of Interface Models

Rana, Gule; Asif, Muhammad; Haider, Nadeem; Bilal, Rubi; Ahsan, Muhammad; Al-Mdallal, Qasem; Jarad, Fahd

doi:https://doi.org/10.1155/2022/1541486

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Numerical Methods for Differential and Integral Equations

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Research Article | Open Access

Volume 2022 | Article ID 1541486 | https://doi.org/10.1155/2022/1541486

A Modified Algorithm Based on Haar Wavelets for the Numerical Simulation of Interface Models

Gule Rana,¹Muhammad Asif,²Nadeem Haider,²Rubi Bilal,¹Muhammad Ahsan,³Qasem Al-Mdallal,⁴and Fahd Jarad^5,6

Academic Editor: Anna Napoli

Received15 Sept 2021

Revised20 May 2022

Accepted13 Jul 2022

Published09 Aug 2022

Abstract

In this paper, a new numerical technique is proposed for the simulations of advection-diffusion-reaction type elliptic and parabolic interface models. The proposed technique comprises of the Haar wavelet collocation method and the finite difference method. In this technique, the spatial derivative is approximated by truncated Haar wavelet series, while for temporal derivative, the finite difference formula is used. The diffusion coefficients, advection coefficients, and reaction coefficients are considered discontinuously across the fixed interface. The newly established numerical technique is applied to both linear and nonlinear benchmark interface models. In the case of linear interface models, Gauss elimination method is used, whereas for nonlinear interface models, the nonlinearity is removed by using the quasi-Newton linearization technique. The errors are calculated for different number of collocation points. The obtained numerical results are compared with the immersed interface method. The stability and convergence of the method are also discussed. On the whole, the numerical results show more efficiency, better accuracy, and simpler applicability of the newly developed numerical technique compared to the existing methods in literature.

1. Introduction

Interface models play an important role in many disciplines including electromagnetic wave propagation, material science, fluid dynamics, and biological systems. The shared boundary between the two intervals in case of one-dimensional domains or between two regions in case of higher-dimensional domains is known as an interface. These domains (intervals or regions) are kept together with the help of suitable jump constraints. These phenomena can be modeled by using partial differential equations (PDEs) or ordinary differential equations (ODEs), where the parameters in these differential equations across the interface separating the two materials or states are discontinuous. Interface model is a mathematical model which considers two identical or different materials at different states having a shared boundary. The example of interface models with same materials in different states is water and ice, while water and oil is an example of interface models with different materials [1, 2]. These models frequently arise in heat conduction, Navier-Stokes flows, crystal growth, wave propagation through nonhomogeneous media, and models of solidification. Most of the interface model equations consist of highly varying coefficients [1, 3, 4]. The approximations of various physical and biomedical models often consist of highly varying coefficients or heterogeneous ODE or PDE models [5].

The solution of these models is a challenge for many standard numerical methods such as finite element method, finite volume method, and finite difference method. These methods have either poor performance or unable to catch the discontinuity in the solution. Due to the numerous applications of such type of models, several numerical methods have been introduced for the solution of these models with regular and irregular geometries in literature. Some of the numerical methods are immersed boundary method (IBM) [6, 7], immersed interface method (IIM) [1, 8], ghost fluid method (GFM) [9], matched interface and boundary method (MIBM) [10–12], the method based on the integral equations approach [13], and finite element methods [14–17].

Recently, wavelet analysis has got much popularity in the approximation theory. Different wavelets and approximating functions are introduced for approximation purpose. Wavelets have simple and fast algorithms, which result better approximation. Among all these wavelets, Haar wavelet has got great importance due to their simplicity and applications. Haar wavelet contains piecewise constant box functions. The Haar wavelet collocation method (HWCM) got attention of many authors due their simple nature, properties of orthogonality, and compact support. The Haar wavelet contains piecewise constant functions; therefore, complicated models can be approximated very easily using these wavelets. Besides, several types of boundary conditions including local and nonlocal conditions can be utilized. Various applications of HWCM in the approximation theory can be seen in [18–27]. Some of the recent work using Haar wavelets is given in [28–36].

In this article, a new approach based on Haar wavelet and finite difference method is developed for the numerical solution of advection-diffusion-reaction type elliptic and parabolic interface models.

The article is organized as follow. In Section 3, definition of the Haar wavelet and their integrals are presented. In Section 4, construction of the newly proposed numerical method based on Haar wavelet and FDM is given. The convergence and stability analysis of the proposed numerical method are discussed in Sections 5 and 6. In Section 7, numerical validation of the method is given. In the last section, conclusion is presented.

2. Governing Models

2.1. Elliptic Interface Model

Consider the following forms of linear and nonlinear elliptic interface models:

At the interface point , the interval is divided into two subintervals and . The functions involved in Equations (1) and (2) are of the form

The Dirichlet boundary conditions at boundary points and are given by

The following interface conditions are considered at the interface point : where , , , and and , , , and are known functions defined on and , respectively.

2.2. Parabolic Interface Model

The following forms of linear and nonlinear parabolic interface models are considered:

The interface point divides the interval into two subintervals and , where , , and are the same as given in the above elliptic problem. The functions involved in Equations (7) and (8) are of the following form:

Subject to the following initial and Dirichlet boundary conditions points and ,

The following interface conditions are considered at the interface point :

The functions , , , and and , , , and are smooth functions defined on and , respectively.

3. Haar Wavelets

The wavelet of the Haar family over is defined as [37] where

In the above equations, and are integers such that , and . The level of the resolution of the Haar wavelet and the translation parameter are represented by the integers and , respectively. For approximation purposes, we consider a maximal value of the integer . The integer is then called maximal level of resolution. We also define . The equation shows the relation among , , and . The minimal and maximal values of can be obtained from the equation . If , , then minimal value is and the maximal value is . For , we get , which is known as scaling function for Haar wavelet family and is defined as

Any square integrable function over the interval can be expressed as infinite sum of functions of the Haar wavelet family as

For approximation purpose, the above series is truncated to a finite sum in the following manner: where is the maximal resolution defined above. All other members of the Haar family can be obtained from Equation (13) by the process of dilation and translation. The following notations are introduced for Haar integrals:

These integrals can be calculated utilizing Equation (13) and are given below. where . For , we have

4. Numerical Procedure

In this section, formulation of numerical technique both for elliptic and parabolic advection-diffusion-reaction type interface models is discussed. The interval of study is considered to be .

4.1. HWCM for Elliptic Model with Single Interface

In this technique, the higher-order spatial derivative is approximated by truncated Haar series; the approximate expressions for the lower order derivatives and for the unknown function are calculated by integration process. The details of the procedure are given below:

Integrating Equation (21), from to , we get Again integrating from to , we have

Similarly, we can approximate the second function over the second subinterval as follows:

Integrating Equation (24), we get the expressions and as follows:

After substituting the Haar expression, Equations (5) and (6) become

The remaining procedure will be explained separately for both linear and nonlinear cases.

4.1.1. Linear Case

Substituting the values of , , and in Equation (1) and simplifying, we have

Similarly, substituting the values of , , and in Equation (1), we get

The following discrete points are used for single interface problem:

After discretization, we get the subsequent forms of Equations (29) and (30):

Equations (32) and (33) combined with Equations (27) and (28) give a linear system of equations with unknowns , , , and . We can write the above system in matrix form as follows: where

The entries of the matrix are given by

Finally, we obtained the following entries of the matrix :

Equation (34) can be solved by any linear solver in order to get the unknown Haar coefficients. Now utilizing these unknown Haar coefficients in Equations (23) and (26), we can easily obtain the approximate solution of the problem.

4.1.2. Nonlinear Case

In nonlinear case, first we linearize Equation (2) by using the following quasi-Newton linearization technique [38]:

After linearizing Equation (2), substituting the Haar approximations for and its derivatives and then discretizing, we get

Equations (41) and (42) along with Equations (27) and (28) give a linear system of size with the unknown Haar coefficients and the values and . The above linear system can be solved by using any linear solver.

4.2. HWCM for Parabolic Model with Single Interface

This is a parabolic interface model. The time derivative is approximated by using the following forward difference formula:

Now approximating the highest order spatial derivative over the first subinterval by truncated Haar series,

Integrating Equation (44), we get the approximate expressions for and as follows:

Similarly, approximating over the second subinterval as follows,

Integrating Equation (47), we obtain the approximate expressions for and as follows:

Substituting the values of , , , , , and in Equations (11) and (12), the interface conditions imply that

The remaining procedure is explained for linear and nonlinear cases separately in the upcoming section.

4.2.1. Linear Case

Substituting Equations (43) and (46) in Equation (7) and simplifying, we have

Similarly by using Equation (43) and Equations (47)–(49) in Equation (7) and simplifying, we have

The following nodes are defined for interface conditions at :

Discretizing, we get the following systems of linear equations:

Equations (55) and (56) combined with Equations (50) and (51) give a linear system of size with unknown coefficients , , , and . In matrix form, the above system can be written as where and are given in Equations (35) and (37), respectively, and

The entries of the matrix are given by

Finally, we can write the elements of the matrix as follows:

From Equation (57), we get

Solving system (61) by any linear solver, we obtained the values of the unknown Haar coefficients , , , and . By utilizing these unknown Haar coefficients in Equations (46) and (49), we can easily obtain the approximate solution of the problem.

4.2.2. Nonlinear Case

In nonlinear interface models first, we linearize problem (8) by using the following quasi-Newton Linearization technique [38]:

Now substituting the approximate expressions for higher-order derivatives, unknown function , and temporal derivative in the linearized equation and discretizing, we obtain the following systems of equations.

Equations (63) and (64) together with Equations (50) and (51) give a linear system of size . Solving the system by any linear solver, we can get the unknown Haar coefficients. Using these unknown Haar coefficients, we can easily obtain the approximate solution.

5. Convergence

Lemma 1 [39]. Assume that with , , and , and then, .

Lemma 2 [39]. Let be continuous on . Then, the error norm at level satisfies where and , and is a positive real number related to the level resolution of the wavelet given by .

Theorem 3. If is the exact solution and is the approximate solution of Equation (1), the error norm at level resolution is given by

Proof. The error estimate of the proposed method at level resolution is given as which implies that where . Now, Equation (68) can be written as Now, using Lemmas 1 and 2, inequality (69) can be written as in which on further simplification and taking squire root, we get

It is concluded that error norm is inversely proportional to level of the Haar wavelet resolution . Hence, the error of the HWCM decreases as increases, i.e.,

Theorem 4. If is the exact solution and is the approximate solution of Equation (7) and if , where is a positive integer, then the error norm at J-th level resolution is given by

Proof. For , see Theorem 3.
For time derivatives, we have used first-order finite difference approximation in Equation (43), so Hence,

6. Stability

In this section, we study the computational stability of the proposed technique. For this purpose, we have observed the maximum eigenvalues of matrix at every time step, which represent the corresponding Haar weights. All the maximum eigenvalues of matrix stay away from zero (see Figure 1), and this leads to a sufficient condition for the proposed technique to be stable. We can write Equations (1) and (2) in the form where is the operator.

Here, is the next time level and is the previous time level. After introducing the Haar wavelet, Equation (80) can be written as where is the weight Haar matrix for operator and is the identity matrix. If the maximum eigenvalue of is , then from Equation (81), the stability condition will be [23, 24]

Here, is the time step which is always positive, i.e., . We have discussed the following three different cases related to Equation (82).

Case 1. If , then Equation (82) gives which is identically satisfied.

Case 2. If , i.e., , where , then Equation (82) gives The inequality holds because the denominator is greater than the numerator.

Case 3. If , i.e., , then Equation (82) gives which is does not holds as the denominator is smaller than the numerator.

Thus, Equation (82) is valid for Cases 1 and 2, which are verified computationally in Figure 1.

Furthermore, examples (1) and (2) are linear and nonlinear steady problems. Therefore, we have found their eigenvalues and listed them in Tables 1 and 2. From the tables, we can observe that all the eigenvalues lie on the left half of the complex plane. Therefore, systems (34) and (42) are stable, because we have a result that state that “A system will be stable if and only if the real part of all eigenvalues of the matrix lie on the left half of the complex plane” [40].

7. Examples and Discussion

This section is devoted to apply HWCM on some benchmark test models. These problems include parabolic and elliptic advection-diffusion-reaction type linear and nonlinear models with single interface conditions. In nonlinear test models, the quasi-Newton linearization technique given in [41] is utilized. The initial guess for nonlinear elliptic problem is taken and stopped the iterations when the criterion of convergence is satisfied. For calculating experimental convergence rates, we have used the following formula: where is the maximum absolute error at collocation points.

Example 1. Consider the following initial-boundary value linear elliptic interface model equation [41]: with boundary conditions: and interface conditions: The exact solution of the test problem is given by

Example 2. Consider the following initial-boundary value nonlinear elliptic interface problem: with boundary conditions: and interface conditions: The exact solution of the test problem is given by

Example 3. Suppose the following initial-boundary value linear parabolic problem with single interface conditions [41]: subject to the following initial and boundary conditions: and interface conditions: The exact solution of the test problem is given by

Example 4. Consider another initial-boundary value linear parabolic interface problem [41]: with the following initial and boundary conditions: and interface conditions: The exact solution of the test problem is given by

Example 5. Consider another linear parabolic interface model: with the subsequent initial conditions and boundary conditions: and interface conditions: The exact solution of the test problem is given by

Example 6. Consider the following nonlinear parabolic interface model: subject to the following initial and boundary conditions: and interface conditions: The exact solution of the test problem is given by

In this section, some numerical experiments comprising linear and nonlinear elliptic and parabolic advection-diffusion-reaction type interface models have been carried out, in order to check the efficiency and better accuracy of the newly proposed numerical technique for these types of models. First, we have discussed elliptic interface models and then parabolic interface models.

In the first two linear and nonlinear elliptic interface models, the errors are decreased to 10^-6 and 10^-7 even for small number of grid points. It is worth mentioning that more accurate numerical results can be obtained if we increase the number grid points. In Table 1, the absolute errors for distinct collocation points are listed. The graph given in Figure 2 also demonstrates that the proposed technique captures the discontinuity very well, where the other methods failed to do so. The computational rate of convergence of the proposed method is approaching to 2, which is theoretically confirmed by Majak et al. [42, 43]. The obtained results are compared with the immersed interface method from literature. The comparability shows that the newly proposed technique is efficient and more accurate for elliptic type interface models than the existing methods. The newly proposed numerical technique is also tested on advection-diffusion-reaction type parabolic interface models, comprising of three linear and one nonlinear models. The obtained point wise absolute errors are mentioned in Tables 3–6. The numerical results are also demonstrated through D visualization of the graphs listed in Figures 3 and 4. From the aforementioned figures, it is clear that the newly proposed technique handled the jump discontinuity at 0.5 and 0.7 very well. The approximate results are compared with the immersed interface method from the existing literature. The comparability shows that the proposed technique has better accuracy with simple implementation.

(a)

(b)

8. Conclusion

In this article, Haar wavelet collocation technique is utilized to solve interface models comprising advection-diffusion-reaction type elliptic and parabolic models with discontinuous coefficients. The newly proposed numerical technique is applicable to both linear and nonlinear interface models. The errors are decreased up to 10^-6 and 10^-7 for small number of collocation points, which is supposed to be better accuracy for practical problems. The D graphs of the estimated and exact solutions also demonstrate that the newly proposed technique handle the jump discontinuity very well, while the other existing techniques failed to capture it. The stability and convergence of the said numerical technique are also proved in the convergence and stability sections, which made the method more powerful. The obtained results are compared with the immersed interface method. The comparability shows that the newly proposed technique is efficient and has better accuracy than immersed interface method.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Copyright © 2022 Gule Rana et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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