Elsevier

Applied Numerical Mathematics

Volume 161, March 2021, Pages 452-468
Applied Numerical Mathematics

High-order compact schemes for semilinear parabolic moving boundary problems

https://doi.org/10.1016/j.apnum.2020.12.004Get rights and content

Abstract

In this paper, we study high-order compact schemes for semilinear parabolic moving boundary problems. We first convert the original problem into an equivalent one defined on a rectangular region by introducing a linear transformation and the well-known exponential transformation. Next, we derive a compact scheme with fourth-order accuracy in the spatial dimension and second-order accuracy in the temporal dimension. Moreover, we prove that the numerical solutions are convergent strictly in the maximum norm by an energy argument. Extending to two-dimensional semilinear moving boundary problems is also provided. Finally, a series of numerical experiments including linear and semilinear examples are carried out to verify that our schemes have more advantages than the one proposed only for the linear moving boundary problem by Cao and Sun (2010) [6].

Introduction

In recent years, there has been an increasing number of moving boundary problems that occur frequently in physical, chemical, meteorological and biological modeling [1], [8], [17], [22], [24], [26], [28]. For example, the process of tumor growth has been formulated into different mathematical models, which can advance our understanding of cancer research [1], [17], [22]. Besides, the moving boundary problems also arise in the descriptions of the oriented movement of cells [8] and movement of the shoreline in a sedimentary ocean [28] and so on.

In terms of analysis of moving-boundary problems, Caboussat and Rappaz study the well-posedness of a one-dimensional free surface problem and obtain the existence and uniqueness results locally in time [5]. Moreover, Muntean and Böhm deal with a one-dimensional coupled system of semi-linear parabolic equations on the moving boundary and prove a global existence and uniqueness theorem of positive weak solutions [20]. Escher and Matioc re-express the problem of the growth of nonnecrotic tumors as an abstract evolution equation and prove local well-posedness in the small Hölder spaces context [10]. Besides, Sun et al. consider the global existence and uniqueness of a transient solution of a non-autonomous free boundary model for tumor growth [25]. However, as we all know, it is extraordinarily challenging to obtain every analytical solution of a moving boundary problem on account of the complexity of boundaries. Therefore, establishing effective and high-accuracy numerical methods is a must.

To the authors' best knowledge, some numerical methods have been described for the solution of time-dependent partial differential equations with moving boundaries, which can be classified in two ways. On the one hand, based on the moving mesh methods, there are many different numerical algorithms proposed by researchers, which can adapt to different problems with specific features [2], [3], [4], [7], [12], [14]. For example, Baines et al. [2], [3], [4] study moving finite element methods and propose moving mesh finite element algorithm for the adaptive solution of nonlinear diffusion equations with moving boundaries in one and two dimensions. Lee et al. [14] construct a velocity-based method for modeling avascular tumor growth, whose estimated order of convergence is mostly less than or equal to two. Huang et al. establish a moving-mesh method based on the geometric conservation law in [7] and study the stability of three moving-mesh finite-difference schemes in the L norm for one-dimensional linear convection-diffusion equations in [12].

On the other hand, an alternative approach is to rescale the original problem with moving boundary to an equivalent problem on a certain fixed domain [6], [11], [13], [17], [18], [21], [25], [33], [38]. Kovács and Lubich study spatial and temporal discretizations of parabolic differential equations with dynamic boundary conditions in a weak formulation [13]. Besides, numerous finite difference schemes have been applied to solve moving boundary problems numerically [11], [17], [18], [21], [25]. Unfortunately, there is no numerical analysis of the convergence and stability of difference schemes in the above literature. Cao and Sun [6] establish a Crank-Nicolson scheme with the unconditional stability for the linear convection-diffusion equation with moving boundaries and prove that the scheme proposed is convergent in the maximum norm and the order of convergence is O(τ2+h2). Yuan et al. study a three-dimensional moving boundary problem on the compressible miscible (oil and water) displacement by a second-order upwind difference fractional steps scheme applicable to parallel computing [33]. Zheng et al. construct an efficient numerical method to solve a fractional reaction-diffusion model with a moving boundary and obtain the convergence and stability [38].

In terms of the compact difference methods widely used to solve partial differential equations numerically, there are a great number of works in the literature on the numerical solution of partial differential equations. The readers interested in this aspect are referred [9], [15], [16], [19], [29], [30], [31], [32], [34], [35], [36], [37] for details. However, to the authors' best knowledge, there are few works in the literature on the compact schemes with the higher-order convergence for partial differential equations with moving boundary conditions.

In reality, moving-boundary problems include both problems with unknown boundary, which are often called Stefan problems, and prescribed-boundary problems. In this work, we focus on the semilinear convection-diffusion equation with prescribed-boundary condition.(FP){vt=κ2vs2γvs+g(v,s,t),(s,t)QT,v(s,0)=v0(s),sΩ0,v(s,t)=Φ(s,t),sΩt,0<tT, where κ and γ are both positive constants, QT={(s,t)R2,sΩt,t(0,T]}, and the boundary value condition can be rewritten asv(s,t)|sΩt={Φ(xl(t),t),s=xl(t),Φ(xr(t),t),s=xr(t).

(FP) can be regarded as an extension of the linear prescribed-boundary problem in [6]. Here, we assume that the initial value v0 in (FP) is regular enough and the exterior force g has the second-order continuous derivative with respect to the first components in the ϵ0-neighborhood of the solution v, where ϵ0 is a positive constant.

Besides, we assume that the boundary value Φ is piecewise smooth and the following consistency conditions hold: v0(s)=Φ(s,0),sΩ0, i.e., v0(xl(0))=Φ(xl(0),0),v0(xr(0))=Φ(xr(0),0). We suppose that xl(t),xr(t)C2[0,T] and xl(t)<xr(t), for any t[0,T]. And the domain Ωt can be defined as Ωt=(xl(t),xr(t)), which is an interval in R for each t[0,T].

In this paper, we develop a kind of compact scheme with high accuracy and high convergence rate to solve (FP) numerically and carry out the corresponding analysis. Firstly, using a linear transformation and an exponential transformation, we can establish a compact scheme for the equation (FP). Secondly, the unique solvability and the convergence are established. Thirdly, we extend the methodology to establish both a Crank-Nicolson scheme and a compact scheme for the two-dimensional moving boundary problem. Finally, we implement some numerical examples to support our theoretical results.

The content of this paper will be organized as follows. In Section 2, we obtain an equivalent system defined on a rectangular region. In Section 3, we establish a compact scheme with fourth-order accuracy in the spatial dimension and second-order accuracy in the temporal dimension, see Theorem 4.5. Besides, unique solvability and convergence of the established scheme are studied in Section 4. Furthermore, a Crank-Nicolson scheme and a compact scheme for a two-dimensional semilinear moving boundary problem are given as well in Section 5. Finally, some results of numerical examples are given in Section 6.

Section snippets

Problem reformation by a linear variable transformation and an exponential transformation

First and foremost, we introduce a linear transformation{s=(1x)xl(t)+xxr(t),0x1,t=t,0tT. Denote w(x,t)=v((1x)xl(t)+xxr(t),t),g˜(w,x,t)=g(v,s,t) andα(t)=κsx2=κ(xr(t)xl(t))2,β(x,t)=stsxγsx=(1x)xl(t)+xxr(t)γxr(t)xl(t). We can obtainwt=α(t)2wx2+β(x,t)wx+g˜(w,x,t),0<x<1,0<tT,w(x,0)=v0(s,0),0x1,w(0,t)=Φ(xl(t),t),w(1,t)=Φ(xr(t),t),0<tT, where v0(s,0)=v0((1x)xl(0)+xxr(0),0).

Furthermore, the exponential transformation [15]w(x,t)=exp(0xβ(ξ,t)2α(t)dξ)u(x,t) is introduced as

Derivation of a compact scheme

Before the presentation of the compact scheme, the uniform-grid discretization is used. Let Ωh={xi|0im} be a uniform mesh on the interval [0,1] with h=1/m, where m is a positive integer and xi=ih,i=0,1,,m. Let Ωτ={tk|0kn} be a uniform mesh on the interval [0,T] with τ=T/n, where n is a positive integer and tk=kτ,k=0,1,,n. Therefore, the rectangular region [0,1]×[0,T] is subdivided by Ωh×Ωτ. Denote Ωhτ=Ωh×Ωτ={(xi,tk)|xi=ih,tk=kτ,0im,0kn}. Lettk+12=12(tk+tk+1),sik=(1xi)xl(tk)+xixr(tk).

Convergence of the compact scheme

In this section, we analyze the solvability and convergence of the compact scheme (CS). Denote Vh={v|v=(v0,v1,,vm)} and introduce the following notations for vVh:v=max0im|vi|,v=h2v02+hi=1m1vi2+h2vm2,|v|1=hi=0m1(δxvi+12)2.

The following lemmas are necessary for our main results.

Lemma 4.1

[23], [27] If vk=(v0k,v1k,,vmk) is a grid function defined on Ωhτ and satisfy v0k=vmk=0, then|vk|12=hi=1m1(δx2vik)vik,vk12|vk|1,vk16|vk|1,i=1m1(Avik)2i=1m1(vik)2.

Proof

For the last inequality, we

Extension to two-dimensional semilinear moving boundary problem

Considering the two-dimensional semilinear moving boundary problem asvt=κ12vs2+κ22vr2γ1vsγ2vr+g(v,s,r,t),(s,r,t)RT, along with the initial conditionv(s,r,0)=v0(s,r),(s,r)Ω1(0)×Ω2(0), and moving boundary conditionv(s,r,t)=Φ(s,r,t),(s,r)(Ω1(t)×Ω2(t)),0<tT, where κ1,κ2,γ1,γ2 are positive constants. RT={(s,r,t)R3,sΩ1(t),rΩ2(t),t(0,T]}, and Ω1(t)=(xl(t),xr(t)),Ω2(t)=(yl(t),yr(t)) can be defined as two domains, where xl(t),xr(t),yl(t),yr(t)C2[0,T] and xl(t)<xr(t),yl(t)<yr(t),

Numerical examples

To verify our results in the previous sections, we implement four numerical examples, including one-dimensional linear problem (Case 1 and Case 2), one-dimensional semilinear problem (Case 3) and two-dimensional problem (Case 4). Although the compact scheme (CS) is established for semilinear parabolic moving boundary problems, it should be used to solve linear problems as well. That is to say that the compact scheme (CS) should apply equally when the equation (FP) reduces to a linear problem.

Conclusion

In summary, we establish the compact scheme for one-dimensional semilinear parabolic moving boundary problems and study the convergence. The proposed method has the convergence order of two in the temporal direction and the convergence order of four in the spatial direction in L-norm, respectively. Furthermore, we extend the methodology to two-dimensional semilinear problems as well in this paper. With the help of the exponential transformation, we can avoid discussing a compact scheme on

Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grant No. 11871435, 11471287, 11501514), the Natural Sciences Foundation of Zhejiang Province (Grant No. LY19A010026), Zhejiang Province “Yucai” Project (2019), Fundamental Research Funds of Zhejiang Sci-Tech University (Grant No. 2019Q072) and Zhejiang Provincial Department of Education Research Project (Grant No. Y201942553).

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