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

Authors: Gule Rana; Muhammad Asif; Nadeem Haider; Rubi Bilal; Muhammad Ahsan; Qasem Al-Mdallal; Fahd Jarad
Journal: Journal of Function Spaces
 (2022)

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## 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. TheL∞ 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.


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## Body

## 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 Section3, 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:(1)αηvηη−βηvηηη+σηvη=fη,a<η<b,(2)ψvηηη,vηη,vη,αη,βη,ση,η=fη,a<η<b.At the interface pointη=ζ, the interval I=a,b is divided into two subintervals I1=a,ζ and I2=ζ,b. The functions involved in Equations (1) and (2) are of the form
(3)αη,βη,vη,ση,fη=α1η,β1η,v1η,σ1η,f1η,forη∈I1,α2η,β2η,v2η,σ2η,f2η,forη∈I2.The Dirichlet boundary conditions at boundary pointsη=a and η=b are given by
(4)va=γ1,vb=γ2.The following interface conditions are considered at the interface pointη=ζ:
(5)vζ=v2ζ−v1ζ=μ1,(6)α−βvηζ=α2−β2ζv2ηζ−α1−β1ζv1ηζ=μ2,where f1η, α1η>0, β1η>0, and σ1η≥0 and f2η, α2η>0, β2η>0, and σ2η≥0 are known functions defined on I1 and I2, respectively.


### 2.2. Parabolic Interface Model

The following forms of linear and nonlinear parabolic interface models are considered:(7)vtη,t+αη,tvηη,t=βη,tvηη,tη−ση,tvη,t+fη,t,a<η<b,0≤t≤1,(8)ψvtη,t,vηηη,t,vηη,t,vη,t,αη,t,βη,t,ση,t,η,t=fη,t,a<η<b,0≤t≤1.The interface pointη=ζ divides the interval I into two subintervals I1 and I2, where I, I1, and I2 are the same as given in the above elliptic problem. The functions involved in Equations (7) and (8) are of the following form:
(9)αη,t,βη,t,vη,t,ση,t,fη,t=α1η,t,β1η,t,v1η,t,σ1η,t,f1η,t,forη∈I1,α2η,t,β2η,t,v2η,t,σ2η,t,f2η,t,forη∈I2.Subject to the following initial and Dirichlet boundary conditions pointsη=a and η=b,
(10)vη,0=v0η,onI,va,t=γ1t,vb,t=γ2t.The following interface conditions are considered at the interface pointη=ζ:
(11)vζ=v2ζ,t−v1ζ,t=μ1t,(12)α−βvηζ=α2ζ,t−β2ζ,tv2ηζ,t−α1ζ,t−β1ζ,tv1ηζ,t=μ2t.The functionsf1η, α1η>0, β1η>0, and σ1η≥0 and f2η, α2η>0, β2η>0, and σ2η≥0 are smooth functions defined on I1 and I2, respectively.


## 2.1. Elliptic Interface Model

Consider the following forms of linear and nonlinear elliptic interface models:(1)αηvηη−βηvηηη+σηvη=fη,a<η<b,(2)ψvηηη,vηη,vη,αη,βη,ση,η=fη,a<η<b.At the interface pointη=ζ, the interval I=a,b is divided into two subintervals I1=a,ζ and I2=ζ,b. The functions involved in Equations (1) and (2) are of the form
(3)αη,βη,vη,ση,fη=α1η,β1η,v1η,σ1η,f1η,forη∈I1,α2η,β2η,v2η,σ2η,f2η,forη∈I2.The Dirichlet boundary conditions at boundary pointsη=a and η=b are given by
(4)va=γ1,vb=γ2.The following interface conditions are considered at the interface pointη=ζ:
(5)vζ=v2ζ−v1ζ=μ1,(6)α−βvηζ=α2−β2ζv2ηζ−α1−β1ζv1ηζ=μ2,where f1η, α1η>0, β1η>0, and σ1η≥0 and f2η, α2η>0, β2η>0, and σ2η≥0 are known functions defined on I1 and I2, respectively.


## 2.2. Parabolic Interface Model

The following forms of linear and nonlinear parabolic interface models are considered:(7)vtη,t+αη,tvηη,t=βη,tvηη,tη−ση,tvη,t+fη,t,a<η<b,0≤t≤1,(8)ψvtη,t,vηηη,t,vηη,t,vη,t,αη,t,βη,t,ση,t,η,t=fη,t,a<η<b,0≤t≤1.The interface pointη=ζ divides the interval I into two subintervals I1 and I2, where I, I1, and I2 are the same as given in the above elliptic problem. The functions involved in Equations (7) and (8) are of the following form:
(9)αη,t,βη,t,vη,t,ση,t,fη,t=α1η,t,β1η,t,v1η,t,σ1η,t,f1η,t,forη∈I1,α2η,t,β2η,t,v2η,t,σ2η,t,f2η,t,forη∈I2.Subject to the following initial and Dirichlet boundary conditions pointsη=a and η=b,
(10)vη,0=v0η,onI,va,t=γ1t,vb,t=γ2t.The following interface conditions are considered at the interface pointη=ζ:
(11)vζ=v2ζ,t−v1ζ,t=μ1t,(12)α−βvηζ=α2ζ,t−β2ζ,tv2ηζ,t−α1ζ,t−β1ζ,tv1ηζ,t=μ2t.The functionsf1η, α1η>0, β1η>0, and σ1η≥0 and f2η, α2η>0, β2η>0, and σ2η≥0 are smooth functions defined on I1 and I2, respectively.


## 3. Haar Wavelets

Theith wavelet of the Haar family over 0,1 is defined as [37]
(13)hiη=1forη∈δ1,δ2,−1forη∈δ2,δ3,0elsewhere,i=2,3,⋯,where
(14)δ1=km,δ2=k+0.5m,δ3=k+1m.In the above equations,m and k are integers such that m=2j,j=0,1,⋯, and k=0,1,⋯,m−1. The level of the resolution of the Haar wavelet and the translation parameter are represented by the integers j and k, respectively. For approximation purposes, we consider a maximal value J of the integer j. The integer J is then called maximal level of resolution. We also define M=2J. The equation i=m+k+1 shows the relation among i, m, and k. The minimal and maximal values of i can be obtained from the equation i=m+k+1. If m=1, k=0, then minimal value is i=2 and the maximal value is i=2M=2J+1. For i=1, we get h1η, which is known as scaling function for Haar wavelet family and is defined as
(15)hiη=1forη∈0,1,0elsewhere.Any square integrable functionfη over the interval 0,1 can be expressed as infinite sum of functions of the Haar wavelet family as
(16)fη=∑i=1∞aihiη.For approximation purpose, the above series is truncated to a finite sum in the following manner:(17)fη=∑i=12Maihiη,where M 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:
(18)Pi,1η=∫0ηhizdz,Pi,ν+1η=∫0ηpi,νzdz,ν=1,2,⋯.These integrals can be calculated utilizing Equation (13) and are given below.
(19)Pi,nη0forη∈0,δ1,1n!η−δ1nforη∈δ1,δ2,1n!η−δ1n−2η−δ2nforη∈δ2,δ3,1n!η−δ1n−2η−δ2n+η−δ3nforη∈δ3,1,n=1,2,.⋯where i=2,3,⋯. For i=1, we have
(20)P1,nη=ηnn!,n=1,2,3,⋯.


## 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 bea,b=0,1.


### 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:(21)v1ηηη=∑i=12Maihiη,η∈I1.Integrating Equation (21), from η to ζ, we get
(22)v1η=v1ηζ+∑i=12Maipi,1η−pi,1ζ,η∈I1.Again integrating from 0 to η, we have
(23)v1η=γ1+ηv1ηζ+∑i=12Maipi,2η−ηpi,1ζ,η∈I1.Similarly, we can approximate the second functionv2η over the second subinterval I2=ζ,1 as follows:
(24)v2ηηη=∑i=12Mbihiη,η∈I2.Integrating Equation (24), we get the expressions v2η and v2ηη as follows:
(25)v2ηη=v2ηζ+∑i=12Mbipi,1η,η∈I2,(26)v2η=γ2−1−ηv2ηζ+∑i=12Mbipi,2η−pi,21,η∈I2.After substituting the Haar expression, Equations (5) and (6) become
(27)γ2−1−ζv2ηζ+∑i=12Mbipi,2ζ−pi,21−γ1+ζv1ηζ+∑i=12Maipi,2ζ−ζpi,1ζ=μ1,(28)α2ζ−β2ζv2ηζ+∑i=12Mbipi,1ζ−α1ζ−β1ζv1ηζ+∑i=12Maipi,1ζ−pi,1ζ=μ2.The remaining procedure will be explained separately for both linear and nonlinear cases.


#### 4.1.1. Linear Case

Substituting the values ofv1η, v1ηη, and v1ηηη in Equation (1) and simplifying, we have
(29)α1η−β1ηηv1ηζ+∑i=12Maipi,1η−pi,1ζ−β1η∑i=12Maihiη+σ1ηγ1+ηv1ηζ+∑i=12Maipi,2η−ηpi,1ζ=f1η,η∈I1.Similarly, substituting the values ofv2η, v2ηη, and v2ηηη in Equation (1), we get
(30)α2η−β2ηηv2ηζ+∑i=12Mbipi,1η−β2η∑i=12Mbihiη+σ2ηγ2−1−ηv2ηζ+∑i=12Mbipi,2η−pi,21=f2η,η∈I2.The following discrete points are used for single interface problem:(31)ηc′=ζc′−0.52M,forc′=1,2,⋯,2M,ζ+1−ζc′−2M−0.52M,forc′=2M+1,2M+2,⋯,4M.After discretization, we get the subsequent forms of Equations (29) and (30):
(32)∑i=12Maiα1ηj−β1ηηjpi,1ηj−pi,1ζ−β1ηjhiηj+σ1ηjpi,2ηj−ηjpi,1ζ+α1ηj−β1ηηj+σ1ηjηju1ηζ=f1ηj−γ1σ1ηj,j=1,2,⋯,2M,(33)∑i=12Mbiα2ηj−β2ηηjpi,1ηj−β2ηjhiηj+σ2ηjpi,2ηj−pi,21+α2ηj−β2ηηj−σ2ηj1−ηju2ηζ=f2ηj−γ2σ2ηj,j=2M+1,2M+2,⋯,4M.Equations (32) and (33) combined with Equations (27) and (28) give a linear system of 4M+2 equations with 4M+2 unknowns ai,i=1,2,⋯,2M, bi,i=1,2,⋯,2M, v1ηζ, and v2ηζ. We can write the above system in matrix form as follows:
(34)SX=Q,where
(35)S=s11⋯s1,2M0⋯0s1,4M+10⋮⋮⋮⋮⋮⋮⋮⋮s2M,1⋯s2M,2M0⋯0s2M,4M+100⋯0s2M+1,2M+1⋯s2M+1,4M0s2M+1,4M+2⋮⋮⋮⋮⋮⋮⋮⋮0⋯0s4M,2M+1⋯s4M,4M0s4M,4M+2s4M+1,1⋯s4M+1,2Ms4M+1,2M+1⋯s4M+1,4Ms4M+1,4M+1s4M+1,4M+2s4M+2,1⋯s4M+2,2Ms4M+2,2M+1⋯s4M+2,4Ms4M+2,4M+1s4M+2,4M+2,(36)X=a1,a2,⋯,a2M,b1,b2,⋯,b2M,v1ηζ,v2ηζT,(37)Q=q1,q2,⋯q4M+2.T.The entries of the matrixS are given by
(38)sj,i=α1ηj−β1ηηjpi,1ηj−pi,1ζ−β1ηjhiηj+σ1ηjpi,2ηj−ηjpi,1ζ1≤i,j≤2M,sj,i=α2ηj−β2ηηjpi,1ηj−β2ηjhiηj+σ2ηjpi,2ηj−pi,212M+1≤i,j≤4M,sj,4M+1=α1ηj−β1ηηj+σ1ηjηj,1≤j≤2M,sj,4M+2=α2ηj−β2ηηj−σ2ηj1−ηj,2M+1≤j≤4M,s4M+1,i=−pi,2ζ−ζpi,1ζ,fori=1,2,⋯,2M,pi−2M,2ζ−pi−2M,21,fori=2M+1,2M+2,⋯,4M,−ζ,fori=4M+1,−1−ζ,fori=4M+2,s4M+2,i=β1ζpi,1ζ−pi,1ζ,fori=1,2,⋯,2M,−β2ζpi−2M,1ζ,fori=2M+1,2M+2,⋯,4M,β1ζ,fori=4M+1,−β2ζ,fori=4M+2.Finally, we obtained the following entries of the matrixQ:
(39)qj=f1ηj−σ1ηjγ1,forj=1,2,⋯,2M,f2ηj−σ2ηjγ2,forj=2M+1,2M+2,⋯,4M,μ1+γ1−γ2,forj=4M+1,μ2+α1ζ−α2ζ,forj=4M+2.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]:
(40)vdvdηn+1=vndvdηn+1+vn+1dvdηn−vndvdηn.After linearizing Equation (2), substituting the Haar approximations for v and its derivatives and then discretizing, we get
(41)ψ∑i=12Maihiηj,v1ηζ+∑i=12Maipi,1ηj−pi,1ζ,γ1+ηjv1ηζ+∑i=12Maipi,2ηj−ηjpi,1ζ,α1ηj,β1ηj,σ1ηj,ηj=f1ηj,j=1,2,⋯,2M,(42)ψ∑i=12Mbihiηj,v2ηζ+∑i=12Mbipi,1ηj,γ2−1−ηjv2ηζ+∑i=12Mbipi,2ηj−pi,21,α2ηj,β2ηj,σ2ηj,ηj=f2ηj,j=2M+1,2M+2,⋯,4M.Equations (41) and (42) along with Equations (27) and (28) give a linear system of size 4M+2×4M+2 with the unknown Haar coefficients ai,i=1,2,3,⋯.,2M,bi,i=1,2,3,⋯.,2M and the values v1ηζ and v2ηζ. 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:(43)vtη,t=vη,tn+1−vη,tnΔt+OΔt.Now approximating the highest order spatial derivativev1ηηη,t over the first subinterval I1=0,ζ by truncated Haar series,
(44)v1ηηη,t=∑i=12Maihiη,η∈I1.Integrating Equation (44), we get the approximate expressions for v1ηη,t and v1η,t as follows:
(45)v1ηη,t=v1ηζ,t+∑i=12Maipi,1η−pi,1ζ,η∈I1,(46)v1η,t=γ1t+ηv1ηζ,t+∑i=12Maipi,2η−ηpi,1ζ,η∈I1.Similarly, approximatingv2ηηη,t over the second subinterval I2=ζ,1 as follows,
(47)v2ηηη,t=∑i=12Mbihiη,η∈I2.Integrating Equation (47), we obtain the approximate expressions for v2ηη,t and v2η,t as follows:
(48)v2ηη,t=v2ηζ,t+∑i=12Mbipi,1η,η∈I2,(49)v2η,t=γ2t−1−ηv2ηζ,t+∑i=12Mbipi,2η−pi,21,η∈I2.Substituting the values ofv1ζ,t, v2ζ,t, α1ζ,t, α2ζ,t, β1ζ,t, and β2ζ,t in Equations (11) and (12), the interface conditions imply that
(50)γ2t−1−ζv2ηζ,t+∑i=12Mbipi,2ζ−pi,21−γ1t+ζv1ηζ,t+∑i=12Maipi,2ζ−ζpi,1ζ=μ1t,(51)α2ζ,t−β2ζ,tv2ηζ,t+∑i=12Mbipi,1ζ−α1ζ,t−β1ζ,tv1ηζ,t+∑i=12Maipi,1ζ−pi,1ζ=μ2t.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
(52)1+Δtσ1η,tγ1t+ηv1ηζ,t+∑i=12Maipi,2η−ηpi,1ζ+Δtα1η,t−β1ηη,tv1ηζ,t+∑i=12Maipi,1η−pi,1ζ−Δtβ1η,t∑i=12Maihiη=Δtf1η,t+v10η,η∈I1.Similarly by using Equation (43) and Equations (47)–(49) in Equation (7) and simplifying, we have
(53)1+Δtσ2η,tγ2t−1−ηv2ηζ,t+∑i=12Mbipi,2η−pi,21+Δtα2η,t−β2ηη,tv2ηζ,t+∑i=12Mbipi,1η−Δtβ2η,t∑i=12Mbihiη=Δtf2η,t+v20η,η∈I2.The following nodes are defined for interface conditions atη=ζ:
(54)ηc′=ζc′−0.52M,forc′=1,2,⋯,2M;ζ+1−ζc′−2M−0.52M,forc′=2M+1,2M+2,⋯,4M.Discretizing, we get the following systems of linear equations:(55)∑i=12Mai1+Δtσ1ηj,tpi,2ηj−ηjpi,1ζ+Δtα1ηj,t−β1ηηj,tpi,1ηj−pi,1ζ−Δtβ1ηj,thiηj+1+Δtσ1ηj,tηj+Δtα1ηj,t−β1ηηj,tv1ηζ,t=Δtf1ηj,t+v10t−1+Δtσ1ηj,tγ1t,j=1,2,⋯,2M,(56)∑i=12Mbi1+Δtσ2ηj,tpi,2ηj−pi,21+Δtα2ηj,t−β2ηηj,tpi,1ηj−Δtβ2ηj,thiηj+−1+Δtσ2ηj,t1−ηj+Δtα2ηj,t−β2ηηj,tv2ηζ,t=Δtf2ηj,t+v20η−γ2t1+Δtσ2ηj,t,j=2M+1,2M+2,⋯,4M.Equations (55) and (56) combined with Equations (50) and (51) give a linear system of size 4M+2×4M+2 with 4M+2 unknown coefficients ai,i=1,2,⋯,2M, bi,i=1,2,⋯,2M, v1ηζ,t, and v2ηζ,t. In matrix form, the above system can be written as
(57)SX=Q,where S and Q are given in Equations (35) and (37), respectively, and
(58)X=a1t,a2t,⋯,a2Mt,b1t,b2t,⋯,b2Mt,v1ηζ,t,v2ηζ,tT.The entries of the matrixS are given by
(59)sj,i=1+Δtσ1ηj,tpi,2ηj−ηjpi,1ζ+Δtα1ηj,t−β1ηηj,tpi,1ηj−pi,1ζ−Δtβ1ηj,thiηj,1≤i,j≤2M,sj,i=1+Δtσ2ηj,tpi,2ηj−pi,21+Δtα2ηj,t−β2ηηj,tpi,1ηj−Δtβ2ηj,thiηj,2M+1≤i,j≤4M,sj,4M+1=1+Δtσ1ηj,tηj+Δtα1ηj,t−β1ηηj,t,1≤j≤2M,sj,4M+2=−1+Δtσ2ηj,t1−ηj+Δtα2ηj,t−β2ηηj,t,2M+1≤j≤4M,s4M+1,i=−pi,2ζ−ζpi,1ζ,fori=1,2,⋯,2M,pi−2M,2ζ−pi−2M,21,fori=2M+1,2M+2,⋯,4M,−ζ,fori=4M+1,−1−ζ,fori=4M+2,s4M+2,i=β1ζ,tpi,1ζ−pi,1ζ,fori=1,2,⋯,2M,−β2ζ,tpi−2M,1ζ,fori=2M+1,2M+2,⋯,4M,β1ζ,t,fori=4M+1,−β2ζ,t,fori=4M+2.Finally, we can write the elements of the matrixQ as follows:
(60)qj=Δtf1ηj,t+v10η−1+Δtσ1ηj,tγ1t,forj=1,2,⋯,2M,Δtf2ηj,t+v20η−1+Δtσ2ηj,tγ2t,forj=2M+1,2M+2,⋯,4M,μ1ζ,t+γ1t−γ2t,forj=4M+1,μ2ζ,t+α1ζ,t−α2ζ,t,forj=4M+2.From Equation (57), we get
(61)X=S−1Q.Solving system (61) by any linear solver, we obtained the values of the unknown Haar coefficients ai,i=1,2,⋯,2M, bi,i=1,2,⋯,2M, v1ηζ,t, and v2ηζ,t. 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]:
(62)v∂v∂ηn+1=vn∂v∂ηn+1+vn+1∂v∂ηn−vn∂v∂ηn.Now substituting the approximate expressions for higher-order derivatives, unknown functionv, and temporal derivative in the linearized equation and discretizing, we obtain the following systems of equations.
(63)ψv1tζ,t,∑i=12Maihiηj,v1ηζ,t+∑i=12Maipi,1ηj−pi,1ζ,γ1t+ηjv1ηζ,t+∑i=12Maipi,2ηj−ηjpi,1ζ,α1ηj,t,β1ηj,t,σ1ηj,t,ηj,t=f1ηj,t,j=1,2,⋯,2M,(64)ψv2tζ,t,∑i=12Mbihiηj,v2ηζ,t+∑i=12Mbipi,1ηj,γ2t−1−ηjv2ηζ,t+∑i=12Mbipi,2ηj−pi,21,α2ηj,t,β2ηj,t,σ2ηj,t,ηj,t=f2ηj,t,j=2M+1,2M+2,⋯,4M.Equations (63) and (64) together with Equations (50) and (51) give a linear system of size 4M+2×4M+2. 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.


## 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:(21)v1ηηη=∑i=12Maihiη,η∈I1.Integrating Equation (21), from η to ζ, we get
(22)v1η=v1ηζ+∑i=12Maipi,1η−pi,1ζ,η∈I1.Again integrating from 0 to η, we have
(23)v1η=γ1+ηv1ηζ+∑i=12Maipi,2η−ηpi,1ζ,η∈I1.Similarly, we can approximate the second functionv2η over the second subinterval I2=ζ,1 as follows:
(24)v2ηηη=∑i=12Mbihiη,η∈I2.Integrating Equation (24), we get the expressions v2η and v2ηη as follows:
(25)v2ηη=v2ηζ+∑i=12Mbipi,1η,η∈I2,(26)v2η=γ2−1−ηv2ηζ+∑i=12Mbipi,2η−pi,21,η∈I2.After substituting the Haar expression, Equations (5) and (6) become
(27)γ2−1−ζv2ηζ+∑i=12Mbipi,2ζ−pi,21−γ1+ζv1ηζ+∑i=12Maipi,2ζ−ζpi,1ζ=μ1,(28)α2ζ−β2ζv2ηζ+∑i=12Mbipi,1ζ−α1ζ−β1ζv1ηζ+∑i=12Maipi,1ζ−pi,1ζ=μ2.The remaining procedure will be explained separately for both linear and nonlinear cases.


### 4.1.1. Linear Case

Substituting the values ofv1η, v1ηη, and v1ηηη in Equation (1) and simplifying, we have
(29)α1η−β1ηηv1ηζ+∑i=12Maipi,1η−pi,1ζ−β1η∑i=12Maihiη+σ1ηγ1+ηv1ηζ+∑i=12Maipi,2η−ηpi,1ζ=f1η,η∈I1.Similarly, substituting the values ofv2η, v2ηη, and v2ηηη in Equation (1), we get
(30)α2η−β2ηηv2ηζ+∑i=12Mbipi,1η−β2η∑i=12Mbihiη+σ2ηγ2−1−ηv2ηζ+∑i=12Mbipi,2η−pi,21=f2η,η∈I2.The following discrete points are used for single interface problem:(31)ηc′=ζc′−0.52M,forc′=1,2,⋯,2M,ζ+1−ζc′−2M−0.52M,forc′=2M+1,2M+2,⋯,4M.After discretization, we get the subsequent forms of Equations (29) and (30):
(32)∑i=12Maiα1ηj−β1ηηjpi,1ηj−pi,1ζ−β1ηjhiηj+σ1ηjpi,2ηj−ηjpi,1ζ+α1ηj−β1ηηj+σ1ηjηju1ηζ=f1ηj−γ1σ1ηj,j=1,2,⋯,2M,(33)∑i=12Mbiα2ηj−β2ηηjpi,1ηj−β2ηjhiηj+σ2ηjpi,2ηj−pi,21+α2ηj−β2ηηj−σ2ηj1−ηju2ηζ=f2ηj−γ2σ2ηj,j=2M+1,2M+2,⋯,4M.Equations (32) and (33) combined with Equations (27) and (28) give a linear system of 4M+2 equations with 4M+2 unknowns ai,i=1,2,⋯,2M, bi,i=1,2,⋯,2M, v1ηζ, and v2ηζ. We can write the above system in matrix form as follows:
(34)SX=Q,where
(35)S=s11⋯s1,2M0⋯0s1,4M+10⋮⋮⋮⋮⋮⋮⋮⋮s2M,1⋯s2M,2M0⋯0s2M,4M+100⋯0s2M+1,2M+1⋯s2M+1,4M0s2M+1,4M+2⋮⋮⋮⋮⋮⋮⋮⋮0⋯0s4M,2M+1⋯s4M,4M0s4M,4M+2s4M+1,1⋯s4M+1,2Ms4M+1,2M+1⋯s4M+1,4Ms4M+1,4M+1s4M+1,4M+2s4M+2,1⋯s4M+2,2Ms4M+2,2M+1⋯s4M+2,4Ms4M+2,4M+1s4M+2,4M+2,(36)X=a1,a2,⋯,a2M,b1,b2,⋯,b2M,v1ηζ,v2ηζT,(37)Q=q1,q2,⋯q4M+2.T.The entries of the matrixS are given by
(38)sj,i=α1ηj−β1ηηjpi,1ηj−pi,1ζ−β1ηjhiηj+σ1ηjpi,2ηj−ηjpi,1ζ1≤i,j≤2M,sj,i=α2ηj−β2ηηjpi,1ηj−β2ηjhiηj+σ2ηjpi,2ηj−pi,212M+1≤i,j≤4M,sj,4M+1=α1ηj−β1ηηj+σ1ηjηj,1≤j≤2M,sj,4M+2=α2ηj−β2ηηj−σ2ηj1−ηj,2M+1≤j≤4M,s4M+1,i=−pi,2ζ−ζpi,1ζ,fori=1,2,⋯,2M,pi−2M,2ζ−pi−2M,21,fori=2M+1,2M+2,⋯,4M,−ζ,fori=4M+1,−1−ζ,fori=4M+2,s4M+2,i=β1ζpi,1ζ−pi,1ζ,fori=1,2,⋯,2M,−β2ζpi−2M,1ζ,fori=2M+1,2M+2,⋯,4M,β1ζ,fori=4M+1,−β2ζ,fori=4M+2.Finally, we obtained the following entries of the matrixQ:
(39)qj=f1ηj−σ1ηjγ1,forj=1,2,⋯,2M,f2ηj−σ2ηjγ2,forj=2M+1,2M+2,⋯,4M,μ1+γ1−γ2,forj=4M+1,μ2+α1ζ−α2ζ,forj=4M+2.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]:
(40)vdvdηn+1=vndvdηn+1+vn+1dvdηn−vndvdηn.After linearizing Equation (2), substituting the Haar approximations for v and its derivatives and then discretizing, we get
(41)ψ∑i=12Maihiηj,v1ηζ+∑i=12Maipi,1ηj−pi,1ζ,γ1+ηjv1ηζ+∑i=12Maipi,2ηj−ηjpi,1ζ,α1ηj,β1ηj,σ1ηj,ηj=f1ηj,j=1,2,⋯,2M,(42)ψ∑i=12Mbihiηj,v2ηζ+∑i=12Mbipi,1ηj,γ2−1−ηjv2ηζ+∑i=12Mbipi,2ηj−pi,21,α2ηj,β2ηj,σ2ηj,ηj=f2ηj,j=2M+1,2M+2,⋯,4M.Equations (41) and (42) along with Equations (27) and (28) give a linear system of size 4M+2×4M+2 with the unknown Haar coefficients ai,i=1,2,3,⋯.,2M,bi,i=1,2,3,⋯.,2M and the values v1ηζ and v2ηζ. The above linear system can be solved by using any linear solver.


## 4.1.1. Linear Case

Substituting the values ofv1η, v1ηη, and v1ηηη in Equation (1) and simplifying, we have
(29)α1η−β1ηηv1ηζ+∑i=12Maipi,1η−pi,1ζ−β1η∑i=12Maihiη+σ1ηγ1+ηv1ηζ+∑i=12Maipi,2η−ηpi,1ζ=f1η,η∈I1.Similarly, substituting the values ofv2η, v2ηη, and v2ηηη in Equation (1), we get
(30)α2η−β2ηηv2ηζ+∑i=12Mbipi,1η−β2η∑i=12Mbihiη+σ2ηγ2−1−ηv2ηζ+∑i=12Mbipi,2η−pi,21=f2η,η∈I2.The following discrete points are used for single interface problem:(31)ηc′=ζc′−0.52M,forc′=1,2,⋯,2M,ζ+1−ζc′−2M−0.52M,forc′=2M+1,2M+2,⋯,4M.After discretization, we get the subsequent forms of Equations (29) and (30):
(32)∑i=12Maiα1ηj−β1ηηjpi,1ηj−pi,1ζ−β1ηjhiηj+σ1ηjpi,2ηj−ηjpi,1ζ+α1ηj−β1ηηj+σ1ηjηju1ηζ=f1ηj−γ1σ1ηj,j=1,2,⋯,2M,(33)∑i=12Mbiα2ηj−β2ηηjpi,1ηj−β2ηjhiηj+σ2ηjpi,2ηj−pi,21+α2ηj−β2ηηj−σ2ηj1−ηju2ηζ=f2ηj−γ2σ2ηj,j=2M+1,2M+2,⋯,4M.Equations (32) and (33) combined with Equations (27) and (28) give a linear system of 4M+2 equations with 4M+2 unknowns ai,i=1,2,⋯,2M, bi,i=1,2,⋯,2M, v1ηζ, and v2ηζ. We can write the above system in matrix form as follows:
(34)SX=Q,where
(35)S=s11⋯s1,2M0⋯0s1,4M+10⋮⋮⋮⋮⋮⋮⋮⋮s2M,1⋯s2M,2M0⋯0s2M,4M+100⋯0s2M+1,2M+1⋯s2M+1,4M0s2M+1,4M+2⋮⋮⋮⋮⋮⋮⋮⋮0⋯0s4M,2M+1⋯s4M,4M0s4M,4M+2s4M+1,1⋯s4M+1,2Ms4M+1,2M+1⋯s4M+1,4Ms4M+1,4M+1s4M+1,4M+2s4M+2,1⋯s4M+2,2Ms4M+2,2M+1⋯s4M+2,4Ms4M+2,4M+1s4M+2,4M+2,(36)X=a1,a2,⋯,a2M,b1,b2,⋯,b2M,v1ηζ,v2ηζT,(37)Q=q1,q2,⋯q4M+2.T.The entries of the matrixS are given by
(38)sj,i=α1ηj−β1ηηjpi,1ηj−pi,1ζ−β1ηjhiηj+σ1ηjpi,2ηj−ηjpi,1ζ1≤i,j≤2M,sj,i=α2ηj−β2ηηjpi,1ηj−β2ηjhiηj+σ2ηjpi,2ηj−pi,212M+1≤i,j≤4M,sj,4M+1=α1ηj−β1ηηj+σ1ηjηj,1≤j≤2M,sj,4M+2=α2ηj−β2ηηj−σ2ηj1−ηj,2M+1≤j≤4M,s4M+1,i=−pi,2ζ−ζpi,1ζ,fori=1,2,⋯,2M,pi−2M,2ζ−pi−2M,21,fori=2M+1,2M+2,⋯,4M,−ζ,fori=4M+1,−1−ζ,fori=4M+2,s4M+2,i=β1ζpi,1ζ−pi,1ζ,fori=1,2,⋯,2M,−β2ζpi−2M,1ζ,fori=2M+1,2M+2,⋯,4M,β1ζ,fori=4M+1,−β2ζ,fori=4M+2.Finally, we obtained the following entries of the matrixQ:
(39)qj=f1ηj−σ1ηjγ1,forj=1,2,⋯,2M,f2ηj−σ2ηjγ2,forj=2M+1,2M+2,⋯,4M,μ1+γ1−γ2,forj=4M+1,μ2+α1ζ−α2ζ,forj=4M+2.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]:
(40)vdvdηn+1=vndvdηn+1+vn+1dvdηn−vndvdηn.After linearizing Equation (2), substituting the Haar approximations for v and its derivatives and then discretizing, we get
(41)ψ∑i=12Maihiηj,v1ηζ+∑i=12Maipi,1ηj−pi,1ζ,γ1+ηjv1ηζ+∑i=12Maipi,2ηj−ηjpi,1ζ,α1ηj,β1ηj,σ1ηj,ηj=f1ηj,j=1,2,⋯,2M,(42)ψ∑i=12Mbihiηj,v2ηζ+∑i=12Mbipi,1ηj,γ2−1−ηjv2ηζ+∑i=12Mbipi,2ηj−pi,21,α2ηj,β2ηj,σ2ηj,ηj=f2ηj,j=2M+1,2M+2,⋯,4M.Equations (41) and (42) along with Equations (27) and (28) give a linear system of size 4M+2×4M+2 with the unknown Haar coefficients ai,i=1,2,3,⋯.,2M,bi,i=1,2,3,⋯.,2M and the values v1ηζ and v2ηζ. 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:(43)vtη,t=vη,tn+1−vη,tnΔt+OΔt.Now approximating the highest order spatial derivativev1ηηη,t over the first subinterval I1=0,ζ by truncated Haar series,
(44)v1ηηη,t=∑i=12Maihiη,η∈I1.Integrating Equation (44), we get the approximate expressions for v1ηη,t and v1η,t as follows:
(45)v1ηη,t=v1ηζ,t+∑i=12Maipi,1η−pi,1ζ,η∈I1,(46)v1η,t=γ1t+ηv1ηζ,t+∑i=12Maipi,2η−ηpi,1ζ,η∈I1.Similarly, approximatingv2ηηη,t over the second subinterval I2=ζ,1 as follows,
(47)v2ηηη,t=∑i=12Mbihiη,η∈I2.Integrating Equation (47), we obtain the approximate expressions for v2ηη,t and v2η,t as follows:
(48)v2ηη,t=v2ηζ,t+∑i=12Mbipi,1η,η∈I2,(49)v2η,t=γ2t−1−ηv2ηζ,t+∑i=12Mbipi,2η−pi,21,η∈I2.Substituting the values ofv1ζ,t, v2ζ,t, α1ζ,t, α2ζ,t, β1ζ,t, and β2ζ,t in Equations (11) and (12), the interface conditions imply that
(50)γ2t−1−ζv2ηζ,t+∑i=12Mbipi,2ζ−pi,21−γ1t+ζv1ηζ,t+∑i=12Maipi,2ζ−ζpi,1ζ=μ1t,(51)α2ζ,t−β2ζ,tv2ηζ,t+∑i=12Mbipi,1ζ−α1ζ,t−β1ζ,tv1ηζ,t+∑i=12Maipi,1ζ−pi,1ζ=μ2t.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
(52)1+Δtσ1η,tγ1t+ηv1ηζ,t+∑i=12Maipi,2η−ηpi,1ζ+Δtα1η,t−β1ηη,tv1ηζ,t+∑i=12Maipi,1η−pi,1ζ−Δtβ1η,t∑i=12Maihiη=Δtf1η,t+v10η,η∈I1.Similarly by using Equation (43) and Equations (47)–(49) in Equation (7) and simplifying, we have
(53)1+Δtσ2η,tγ2t−1−ηv2ηζ,t+∑i=12Mbipi,2η−pi,21+Δtα2η,t−β2ηη,tv2ηζ,t+∑i=12Mbipi,1η−Δtβ2η,t∑i=12Mbihiη=Δtf2η,t+v20η,η∈I2.The following nodes are defined for interface conditions atη=ζ:
(54)ηc′=ζc′−0.52M,forc′=1,2,⋯,2M;ζ+1−ζc′−2M−0.52M,forc′=2M+1,2M+2,⋯,4M.Discretizing, we get the following systems of linear equations:(55)∑i=12Mai1+Δtσ1ηj,tpi,2ηj−ηjpi,1ζ+Δtα1ηj,t−β1ηηj,tpi,1ηj−pi,1ζ−Δtβ1ηj,thiηj+1+Δtσ1ηj,tηj+Δtα1ηj,t−β1ηηj,tv1ηζ,t=Δtf1ηj,t+v10t−1+Δtσ1ηj,tγ1t,j=1,2,⋯,2M,(56)∑i=12Mbi1+Δtσ2ηj,tpi,2ηj−pi,21+Δtα2ηj,t−β2ηηj,tpi,1ηj−Δtβ2ηj,thiηj+−1+Δtσ2ηj,t1−ηj+Δtα2ηj,t−β2ηηj,tv2ηζ,t=Δtf2ηj,t+v20η−γ2t1+Δtσ2ηj,t,j=2M+1,2M+2,⋯,4M.Equations (55) and (56) combined with Equations (50) and (51) give a linear system of size 4M+2×4M+2 with 4M+2 unknown coefficients ai,i=1,2,⋯,2M, bi,i=1,2,⋯,2M, v1ηζ,t, and v2ηζ,t. In matrix form, the above system can be written as
(57)SX=Q,where S and Q are given in Equations (35) and (37), respectively, and
(58)X=a1t,a2t,⋯,a2Mt,b1t,b2t,⋯,b2Mt,v1ηζ,t,v2ηζ,tT.The entries of the matrixS are given by
(59)sj,i=1+Δtσ1ηj,tpi,2ηj−ηjpi,1ζ+Δtα1ηj,t−β1ηηj,tpi,1ηj−pi,1ζ−Δtβ1ηj,thiηj,1≤i,j≤2M,sj,i=1+Δtσ2ηj,tpi,2ηj−pi,21+Δtα2ηj,t−β2ηηj,tpi,1ηj−Δtβ2ηj,thiηj,2M+1≤i,j≤4M,sj,4M+1=1+Δtσ1ηj,tηj+Δtα1ηj,t−β1ηηj,t,1≤j≤2M,sj,4M+2=−1+Δtσ2ηj,t1−ηj+Δtα2ηj,t−β2ηηj,t,2M+1≤j≤4M,s4M+1,i=−pi,2ζ−ζpi,1ζ,fori=1,2,⋯,2M,pi−2M,2ζ−pi−2M,21,fori=2M+1,2M+2,⋯,4M,−ζ,fori=4M+1,−1−ζ,fori=4M+2,s4M+2,i=β1ζ,tpi,1ζ−pi,1ζ,fori=1,2,⋯,2M,−β2ζ,tpi−2M,1ζ,fori=2M+1,2M+2,⋯,4M,β1ζ,t,fori=4M+1,−β2ζ,t,fori=4M+2.Finally, we can write the elements of the matrixQ as follows:
(60)qj=Δtf1ηj,t+v10η−1+Δtσ1ηj,tγ1t,forj=1,2,⋯,2M,Δtf2ηj,t+v20η−1+Δtσ2ηj,tγ2t,forj=2M+1,2M+2,⋯,4M,μ1ζ,t+γ1t−γ2t,forj=4M+1,μ2ζ,t+α1ζ,t−α2ζ,t,forj=4M+2.From Equation (57), we get
(61)X=S−1Q.Solving system (61) by any linear solver, we obtained the values of the unknown Haar coefficients ai,i=1,2,⋯,2M, bi,i=1,2,⋯,2M, v1ηζ,t, and v2ηζ,t. 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]:
(62)v∂v∂ηn+1=vn∂v∂ηn+1+vn+1∂v∂ηn−vn∂v∂ηn.Now substituting the approximate expressions for higher-order derivatives, unknown functionv, and temporal derivative in the linearized equation and discretizing, we obtain the following systems of equations.
(63)ψv1tζ,t,∑i=12Maihiηj,v1ηζ,t+∑i=12Maipi,1ηj−pi,1ζ,γ1t+ηjv1ηζ,t+∑i=12Maipi,2ηj−ηjpi,1ζ,α1ηj,t,β1ηj,t,σ1ηj,t,ηj,t=f1ηj,t,j=1,2,⋯,2M,(64)ψv2tζ,t,∑i=12Mbihiηj,v2ηζ,t+∑i=12Mbipi,1ηj,γ2t−1−ηjv2ηζ,t+∑i=12Mbipi,2ηj−pi,21,α2ηj,t,β2ηj,t,σ2ηj,t,ηj,t=f2ηj,t,j=2M+1,2M+2,⋯,4M.Equations (63) and (64) together with Equations (50) and (51) give a linear system of size 4M+2×4M+2. 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.


## 4.2.1. Linear Case

Substituting Equations (43) and (46) in Equation (7) and simplifying, we have
(52)1+Δtσ1η,tγ1t+ηv1ηζ,t+∑i=12Maipi,2η−ηpi,1ζ+Δtα1η,t−β1ηη,tv1ηζ,t+∑i=12Maipi,1η−pi,1ζ−Δtβ1η,t∑i=12Maihiη=Δtf1η,t+v10η,η∈I1.Similarly by using Equation (43) and Equations (47)–(49) in Equation (7) and simplifying, we have
(53)1+Δtσ2η,tγ2t−1−ηv2ηζ,t+∑i=12Mbipi,2η−pi,21+Δtα2η,t−β2ηη,tv2ηζ,t+∑i=12Mbipi,1η−Δtβ2η,t∑i=12Mbihiη=Δtf2η,t+v20η,η∈I2.The following nodes are defined for interface conditions atη=ζ:
(54)ηc′=ζc′−0.52M,forc′=1,2,⋯,2M;ζ+1−ζc′−2M−0.52M,forc′=2M+1,2M+2,⋯,4M.Discretizing, we get the following systems of linear equations:(55)∑i=12Mai1+Δtσ1ηj,tpi,2ηj−ηjpi,1ζ+Δtα1ηj,t−β1ηηj,tpi,1ηj−pi,1ζ−Δtβ1ηj,thiηj+1+Δtσ1ηj,tηj+Δtα1ηj,t−β1ηηj,tv1ηζ,t=Δtf1ηj,t+v10t−1+Δtσ1ηj,tγ1t,j=1,2,⋯,2M,(56)∑i=12Mbi1+Δtσ2ηj,tpi,2ηj−pi,21+Δtα2ηj,t−β2ηηj,tpi,1ηj−Δtβ2ηj,thiηj+−1+Δtσ2ηj,t1−ηj+Δtα2ηj,t−β2ηηj,tv2ηζ,t=Δtf2ηj,t+v20η−γ2t1+Δtσ2ηj,t,j=2M+1,2M+2,⋯,4M.Equations (55) and (56) combined with Equations (50) and (51) give a linear system of size 4M+2×4M+2 with 4M+2 unknown coefficients ai,i=1,2,⋯,2M, bi,i=1,2,⋯,2M, v1ηζ,t, and v2ηζ,t. In matrix form, the above system can be written as
(57)SX=Q,where S and Q are given in Equations (35) and (37), respectively, and
(58)X=a1t,a2t,⋯,a2Mt,b1t,b2t,⋯,b2Mt,v1ηζ,t,v2ηζ,tT.The entries of the matrixS are given by
(59)sj,i=1+Δtσ1ηj,tpi,2ηj−ηjpi,1ζ+Δtα1ηj,t−β1ηηj,tpi,1ηj−pi,1ζ−Δtβ1ηj,thiηj,1≤i,j≤2M,sj,i=1+Δtσ2ηj,tpi,2ηj−pi,21+Δtα2ηj,t−β2ηηj,tpi,1ηj−Δtβ2ηj,thiηj,2M+1≤i,j≤4M,sj,4M+1=1+Δtσ1ηj,tηj+Δtα1ηj,t−β1ηηj,t,1≤j≤2M,sj,4M+2=−1+Δtσ2ηj,t1−ηj+Δtα2ηj,t−β2ηηj,t,2M+1≤j≤4M,s4M+1,i=−pi,2ζ−ζpi,1ζ,fori=1,2,⋯,2M,pi−2M,2ζ−pi−2M,21,fori=2M+1,2M+2,⋯,4M,−ζ,fori=4M+1,−1−ζ,fori=4M+2,s4M+2,i=β1ζ,tpi,1ζ−pi,1ζ,fori=1,2,⋯,2M,−β2ζ,tpi−2M,1ζ,fori=2M+1,2M+2,⋯,4M,β1ζ,t,fori=4M+1,−β2ζ,t,fori=4M+2.Finally, we can write the elements of the matrixQ as follows:
(60)qj=Δtf1ηj,t+v10η−1+Δtσ1ηj,tγ1t,forj=1,2,⋯,2M,Δtf2ηj,t+v20η−1+Δtσ2ηj,tγ2t,forj=2M+1,2M+2,⋯,4M,μ1ζ,t+γ1t−γ2t,forj=4M+1,μ2ζ,t+α1ζ,t−α2ζ,t,forj=4M+2.From Equation (57), we get
(61)X=S−1Q.Solving system (61) by any linear solver, we obtained the values of the unknown Haar coefficients ai,i=1,2,⋯,2M, bi,i=1,2,⋯,2M, v1ηζ,t, and v2ηζ,t. 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]:
(62)v∂v∂ηn+1=vn∂v∂ηn+1+vn+1∂v∂ηn−vn∂v∂ηn.Now substituting the approximate expressions for higher-order derivatives, unknown functionv, and temporal derivative in the linearized equation and discretizing, we obtain the following systems of equations.
(63)ψv1tζ,t,∑i=12Maihiηj,v1ηζ,t+∑i=12Maipi,1ηj−pi,1ζ,γ1t+ηjv1ηζ,t+∑i=12Maipi,2ηj−ηjpi,1ζ,α1ηj,t,β1ηj,t,σ1ηj,t,ηj,t=f1ηj,t,j=1,2,⋯,2M,(64)ψv2tζ,t,∑i=12Mbihiηj,v2ηζ,t+∑i=12Mbipi,1ηj,γ2t−1−ηjv2ηζ,t+∑i=12Mbipi,2ηj−pi,21,α2ηj,t,β2ηj,t,σ2ηj,t,ηj,t=f2ηj,t,j=2M+1,2M+2,⋯,4M.Equations (63) and (64) together with Equations (50) and (51) give a linear system of size 4M+2×4M+2. 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 thatv∈C2−∞,∞ with v′≤K,∀η∈a,b, K>0, and v=∑i=0∞λihix, and then, λi≤K2−3j−2/2.Lemma 2 [39].
Letv∈C2−∞,∞ be continuous on a,b. Then, the error norm at Jth level satisfies
(65)EJ2≤K2122−2J,where v′≤K,∀η∈a,b and K>0, and M is a positive real number related to the Jth level resolution of the wavelet given by M=2J.Theorem 3.
Ifvη is the exact solution and v2Mη is the approximate solution of Equation (1), the error norm at Jth level resolution is given by
(66)EJη=v−v2M=O2−32J.Proof.
The error estimate of the proposed method atJth level resolution is given as
(67)EJη=v−v2M=∑i=2M+1∞aipi,2η−ηpi,1ζ,which implies that
(68)EJη2=∫−∞∞∑i=2M+1∞aipi,2η−ηpi,1ζ,∑l=2M+1∞alpl,2η−ηpl,1ζdη=∑i=2M+1∞∑l=2M+1∞∫abaialpi,2η−ηpi,1ζpl,2η−ηpl,1ζdη≤∑i=2M+1∞∑l=2M+1∞aialKi,l,where Ki,l=Supi,l∫abpi,2η−ηpi,1ζpl,2η−ηpl,1ζdη. Now, Equation (68) can be written as
(69)EJη2≤∑i=2M+1∞aia2M+1Ki,2M+1+a2M+2Ki,2M+2+⋯≤∑i=2M+1∞aiKia2M+1+a2M+2+⋯,whereKi=SuplKi,l≤∑i=2M+1∞aiKia2M+1+aiKia2M+2+⋯≤∑i=2M+1∞aiKia2M+1+aiKia2M+2+⋯.
Now, using Lemmas1 and 2, inequality (69) can be written as
(70)EJη2≤K2−3.2J+11−2−3/2∑i=2M+1∞aiKi≤K1K2−3.2J+11−2−3/2whereK1=SupKi,in which on further simplification and taking squire root, we get
(71)EJη≤K1K2−3.2J+11−2−3/2≤O2−32J.It is concluded that error norm is inversely proportional to level of the Haar wavelet resolutionJ. Hence, the error of the HWCM decreases as J increases, i.e.,
(72)EJη⟶0asJ⟶∞,⇒EJη⟶0asM⟶∞.Theorem 4.
Ifvη,tp is the exact solution and v2Mη,tp is the approximate solution of Equation (7) and if p=0,1,2⋯P, where P is a positive integer, then the error norm at J-th level resolution is given by
(73)Error=EJη+EJtp=O2−32J+OΔt.Proof.
Forp=0,1,2⋯P,
(74)EJη=O2−32J,see Theorem 3.
For time derivatives, we have used first-order finite difference approximation in Equation (43), so
(75)EJtp=OΔt.
Hence,(76)Error=EJη+EJtp=O2−32J+OΔt.


## 6. Stability

In this section, we study the computational stability of the proposed technique. For this purpose, we have observed the maximum eigenvalues of matrixS at every time step, which represent the corresponding Haar weights. All the maximum eigenvalues of matrix S 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
(77)vt=Lvη,t+fη,t,where L is the operator.
(78)Lvt=∂∂ηBη,t∂∂η−αη,t−δη,tvη,t+fη,t,(79)I−δtLvη,t=vη,t0+δtfη,t,(80)vη,t=I−δtL−1vη,t0+δtI−δtL−1fη,t.Figure 1
The stability analysis of the proposed method for different examples atN=64, Δt=0.01/32, t=1, a=0, and b=1.Here,t is the next time level and t0 is the previous time level. After introducing the Haar wavelet, Equation (80) can be written as
(81)vη,t=I−δtH−1vη,t0+δtI−δtH−1fη,t,where H is the weight Haar matrix for operator L and I is the identity matrix. If the maximum eigenvalue of H is λ, then from Equation (81), the stability condition will be [23, 24]
(82)11−δtλ≤1.Here,δt is the time step which is always positive, i.e., δt>0. We have discussed the following three different cases related to Equation (82).Case 1.
Ifλ=0, then Equation (82) gives
(83)11−δtλ=11=1,which is identically satisfied.Case 2.
Ifλ<0, i.e., λ=−ξ2, where ξ∈ℝ, then Equation (82) gives
(84)11−δtλ=11+δtξ2<1.The inequality holds because the denominator is greater than the numerator.Case 3.
Ifλ>0, i.e., λ=ξ2, then Equation (82) gives
(85)11−δtλ=11−δtξ2>1,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 AX=B will be stable if and only if the real part of all eigenvalues of the matrix A lie on the left half of the complex plane” [40].Table 1
Analysis of errors for Example1.
JNEcN proposed techniqueCPU time in secEigenvaluesNIIMRcN2162.7551×10−40.454092-1.7826206.1×10−31.82753327.3171×10−50.492690-2.1337401.4×10−31.91284641.8859×10−50.582250-2.4974803.2779×10−41.956051284.7873×10−61.096531-2.88111607.9702×10−51.978062561.2060×10−62.806679-3.28623201.9644×10−51.9890Table 2
Analysis of errors for Example2.
JNEcN proposed techniqueCUP time in secEigenvaluesNo. of iterationsRcN187.1012×10−40.009185-3.226442161.9064×10−40.031301-3.226441.89723324.9514×10−50.097082-3.366341.94494641.2626×10−50.392260-3.226441.971451283.1885×10−61.514269-3.366341.985462568.0118×10−76.019626-3.357741.992775122.0081×10−723.903250-3.226441.9962


## 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 0.1 and stopped the iterations when the criterion of convergence 10−5 is satisfied. For calculating experimental convergence rates, we have used the following formula:
(86)RcT=LogEcT/2/EcTLog2,where EcT is the maximum absolute error at T collocation points.Example 1.
Consider the following initial-boundary value linear elliptic interface model equation [41]:
(87)v1ηη−η3v1ηηη+v1η=η33,η∈0,0.5,13v2ηη−ηv2ηηη+2v2η=−8η2+2η3,η∈0.5,1,with boundary conditions:
(88)v10=0,v21=1.and interface conditions:
(89)v=13,α−βvη=−1.
The exact solution of the test problem is given by(90)v1η=η3,η∈0,0.5,v2η=η,η∈0.5,1.Example 2.
Consider the following initial-boundary value nonlinear elliptic interface problem:(91)v1ηη−ηv1ηηη+2v12η=2η2,η∈0,0.5,13v2ηη−2ηv2ηηη+2v22η=e−η2223η−8η3+2e−η2,η∈0.5,1,with boundary conditions:
(92)v10=0,v21=e−1.and interface conditions:
(93)v=e−0.25−12,α−βvη=e−0.25−16.
The exact solution of the test problem is given by(94)v1η=η,η∈0,0.5,v2η=e−η2,η∈0.5,1.Example 3.
Suppose the following initial-boundary value linear parabolic problem with single interface conditions [41]:
(95)v1tη,t+v1ηη,t−η3v1ηη,tη+v1η,t=−η33sint+η33cost,η∈0,0.5,v2tη,t−ηv2ηη,tη=−η3sint+−9η2cost,η∈0.5,1,subject to the following initial and boundary conditions:
(96)v1η,0=η33v10,t=0,v21,t=cost,and interface conditions:
(97)v=2/30.53cost,α−βvη=−8/30.53cost−1.
The exact solution of the test problem is given by(98)v1η,t=η33cost,η∈0,0.5,v2η,t=η3cost,η∈0.5,1.Example 4.
Consider another initial-boundary value linear parabolic interface problem [41]:
(99)v1tη,t+v1ηη,t−η3v1ηη,tη+v1η,t=−η+1sint+53+ηcost,η∈0,0.5,v2tη,t−ηv2ηη,tη=−ηsint−cost,η∈0.5,1,with the following initial and boundary conditions:
(100)v1η,0=η+1v10,t=cost,v21,t=cost,and interface conditions:
(101)v=−cost,α−βvη=−13cost−1.
The exact solution of the test problem is given by(102)v1η,t=η+1cost,η∈0,0.5,v2η,t=ηcost,η∈0.5,1.Example 5.
Consider another linear parabolic interface model:(103)v1tη,t+v1ηη,t−η3v1ηη,tη=23cost−ηsint,η∈0,0.5,v2tη,t+v1ηη,t−ηv2ηη,tη=−η+12sint,η∈0.5,1,with the subsequent initial conditions and boundary conditions:
(104)v1η,0=η,v10,t=0,v21,t=32cost,and interface conditions:
(105)v=12cost,α−βvη=−13cost.
The exact solution of the test problem is given by(106)v1η,t=ηcost,η∈0,0.5,v2η,t=η+12cost,η∈0.5,1.Example 6.
Consider the following nonlinear parabolic interface model:(107)v1tη,t+v1ηη,t−η3v1ηη,tη+v12η,t=−η4e−t−43η3e−t+η8e−2t,η∈0,0.5,v2tη,t−ηv2ηη,tη+v22η,t=−12η4+116e−t−8η3e−t+14η8+η48+1256e−2t,η∈0.5,1,subject to the following initial and boundary conditions:
(108)v1η,0=η4,v10,t=0,v21,t=1732e−t,and interface conditions:
(109)v=0,α−βvη=−e−t24−1.
The exact solution of the test problem is given by(110)v1η,t=η4e−t,η∈0,0.5,v2η,t=12η4+116e−t,η∈0.5,1.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 3D 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.Figure 2
Comparability of exact and estimated solution forN=32 (a) and plot of EcT (b) for Example 1.
(a)(b)Table 3
Analysis of errors for Example3.
JNΔtEcN proposed techniqueCPU time in secNIIM181/81.8394×10−30.27945283.7×10−22161/167.7251×10−40.345287167.1×10−33321/323.5115×10−40.612788321.5×10−34641/641.6775×10−42.966826643.0×10−451281/1288.1976×10−520.0570561281.0×10−4Table 4
Analysis of errors for Example4.
JNΔtEcN proposed techniqueCPU time in secNIIM180.1/81.5×10−30.39592181.02×10−22160.1/167.5054×10−40.748104162.4×10−33320.1/323.7709×10−43.148897326.0×10−44640.1/641.8867×10−425.324517641.0×10−451280.1/1289.4324×10−5194.2371041281.0×10−4Table 5
Analysis of errors for Example5.
JNΔtEcN proposed techniqueCPU time in secNIIM2160.001/163.6548×10−634.662755202.9061×10−53320.001/321.8282×10−6272.116930406.6497×10−64640.001/649.1465×10−72460.769702801.6343×10−651280.001/1284.5753×10−735500.7673791604.0802×10−7Table 6
Analysis of errors Example6.
JNΔtEcN proposed techniqueCPU time in sec180.1/86.0232×10−40.1573022160.1/161.4666×10−41.1470913320.1/323.3345×10−511.1472774640.1/646.8252×10−681.10428751280.1/1281.3904×10−6588.111313Figure 3
Comparability of exact and estimated results for Example3.Figure 4
Comparability of exact and estimated results for Example4.


## 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 3D 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.



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