A recent survey of permutation trinomials over finite fields

1.   Introduction

A class of problems that give rise of singular behavior are elliptic interface problems with imperfect contact ^[1], and it is featured by the implicit jump condition imposed on the imperfect contact interface, and the jumping quantity of the unknown is related to the flux across the interface. The elliptic interface problems with imperfect contact have been applied to model the Stefan problem of the solidification process and crystal growth, composite materials, multi-phase flows ^[2,3] and the problem of temperature discontinuity between the gas and cooling solid surface ^[4]. More examples include the heat conduction between materials of the different heat capacities and conductivities and interface diffusion processes ^[5,6], the temperature discontinuity between a gas and cooling solid surface ^[4], the conjugate heat transfer problem in thermodynamic processes between materials that are thermally coupled through non-adiabatic contacts ^[7], etc.

We consider the one-dimensional (1D) elliptic interface problem with imperfect contact

together with the following implicit jump conditions across the interface $\alpha$

where $0 < \alpha < 1$, $\vec{n}$ is a unit normal to the interface pointing from $\Omega^{-}$ to $\Omega^{+}$. $g^{\pm}$ denotes the right and left limits of the function $g$ at the point $\alpha$, and $[g] = g^{+} - g^{-}$. Without loss of generality, we assume that $\Omega$ is computational domain, and the interface $\alpha$ separates $\Omega$ into two sub-domains $\Omega^{-}$ and $\Omega^{+}$.

The solution of the elliptic interface problems is often discontinuous due to discontinuous coefficients or singular sources across the interface. To recover the numerical accuracy near the interface, a variety of methods have been developed via enforcing the jump conditions (1.2) and (1.1) into numerical discretization such that accurate and robust numerical algorithms can be designed. Finite difference methods (FDMs) constitute a commonly used approach for elliptic interface problems. Since the publication of the original immersed interface method (IIM) ^[8], they have been applied to various problems, such as the Stokes flow with elastic boundaries or surface tension ^[9], incompressible flow based on the Navier-Stokes equations with singular source terms ^[10], and nonlinear problems in magneto-rheological fluids ^[11]. Many further developments and analysis in various aspects of the FDM for elliptic interface problems were carried out to improve the accuracy, stability or efficiency ^{[12,13,14,15,16,17,18,19]}. Many other elegant methods have been proposed in the past decade, including the ghost fluid method ^[20,21], finite volume method ^[7,22,23,24], the matched interface and boundary (MIB) method ^[25,26,27], etc. Finite element methods (FEMs) constitute another common practice to resolve the elliptic interface problems. With the help of body-fitted unstructured meshes, FEMs ^[28,29,30] have been developed for handling elliptic interfaces and irregular geometries. For interfaces with complex topologies, the construction of high quality body-fitted meshes could be difficult and time-consuming. This motivates the development of immersed FEMs based on non-body-fitted structured meshes ^{[31,32,33,34,35,36,37,38]}, etc. Nevertheless, most interface schemes in the literatures are designed to be of second-order accuracy.

The strategy for generating higher-order difference schemes can be roughly divided into two categories. (1) The first category is expanding the stencil and including more points in the schemes ^{[15,16,25,26,27,39,40,41,42]}. Advantages of this approaches is achieving high-order accuracy and using lower order jump conditions only. The obvious disadvantages of such approach is creating large matrix bandwidths and complicating the numerical treatment near the boundaries. Gibou and Fedkiw ^[16] introduced an $O(h^4)$ accurate finite difference discretization for the Laplace and heat equations on an irregular domain. Previous scholars ^[25,26,27] presented high-order MIB methods for solving elliptic equations with discontinuous coefficients and singular sources on Cartesian grids. This type of method is based on the use of fictitious points to achieve high-order accuracy. Zhong ^[40] presented a high-order IIM with general jump conditions by employing fictitious points to achieve high-order accuracy and using lower-order jump conditions only. Similar to the MIB approach, a wide stencil with more points are involved at irregular points. Feng and Zhao ^[42] introduced a new Cartesian grid FDM based on the fourth-order accurate MIB method. Colnago et al. ^[15] presented a high-order IIM for solving Poisson equations with discontinuous coefficients on Cartesian grids, it combines the FDM and ghost node strategy and requires only the ordinary jumps of the function. (2) The second category involves using the differential equation and the interface relations as additional identities ^[13,39,41]. The advantage of this approach is that the scheme is compact and minimal stencil is used. The disadvantage is that to maintain high-order accuracy, it requires the knowledge of jump conditions of high-order derivatives. Li and Ito ^[13] constructed a fourth-order IIM by computing high-order jump conditions involving mixed derivatives. Linnick and Fasel ^[39] presented a high-order IIM for simulating unsteady incompressible flow in an irregular domain. Instead of using analytical jump conditions, they compute the jump conditions for higher derivatives numerically. Angelova and Vulkov ^[41] presented high-order compact FDM for elliptic equations with intersecting interfaces by using the differential equation and the jump (interface) relations as additional identities, which can be differentiated to eliminate higher-order local truncation errors. For FEMs, the higher-order convergence crucially depends on how well the interface is resolved by the triangular mesh. In practice, subparametric, isoparametric or superparametric elements are usually employed to secure the optimal order of $p+1$ in the $L_2$ norm for a polynomial order of $p$, for both continuous ^[33,43] and discontinuous Galerkin ^{[34,35,44,45,46]} FEMs. It is worth pointing out that most of the numerical methods in literatures are aimed at the elliptical interface problem with homogeneous jump conditions $([u] = 0, [\beta u_{\vec{n}}] = 0)$ or nonhomogeneous jump conditions where the jumps of the temperature as well as the conductive heat flux along the interface are known explicitly (say $[u] = g_1$, $[\beta \partial u/ \partial \vec{n}] = g_2$, with known $g_1$ and $g_2$). When the elliptic interface problem has an implicit jump condition as in (1.2), there are comparatively few numerical methods for solving such problems.

In this paper, we consider the 1D elliptic interface problems with imperfect contact, where the implicit jump condition is a setting for the interface and the jump of the primary variable is proportional to the normal flux across the interface. A class of high-order FDMs is constructed for the 1D elliptic interface for both the body-fitted and non-body-fitted mesh. For each case, the second-, third- and fourth-order approximations of the implicit jump condition are provided by using the original variables on both sides of the interface and a set of jump conditions and its high order derivatives. Numerical examples are presented to verify the performance of the scheme. The numerical results show that the presented schemes can reach the theoretical accuracy for solving the elliptic interface problems with imperfect contact.

The rest of this paper is organized as follows. In Section 2, we formulate the scheme for 1D elliptic equations with imperfect interfaces. Section 3 constructs the approximation of the normal derivative for the interface. Section 4 computes the primary variables $u^{+}$ and $u^{-}$ for the interface. Numerical examples are provided in Section 5 to demonstrate the accuracy and stability of the presented scheme. A concluding remark is given in Section 6.

2.   High-order compact difference scheme

We assume that $\beta(x)$ in (1.1) is a piecewise smooth function with a jump at the interface $\alpha$, and that $\beta(x)$ has upper and lower boundries:

where $\beta_{min}$ and $\beta_{max}$ are two constants. The source term $f(x)$ is piecewise smooth.

Introduce a uniform grid $x_i = ih, i = 0, 1, \cdots, n$ with $h = \frac{1}{n}$. The fourth-order compact difference scheme is constructed and three-point computational stencil for each point is adopted. For node $i$, the three points in computational stencil are $\{x_{i-1}, x_i, x_{i+1}\}$. Without loss of generality, assume the interface $\alpha$ is located in a grid interval, $x_k \leq \alpha \leq x_{k+1}$. Because of the interface is located in the stencil, the points $k$ and $k+1$ are named as irregular points, and $i \ne k, k+1$ are name as regular points.

At a regular point $i$, i.e., $i \ne k, k+1$, for convenience, we briefly give the construction process of the fourth order compact difference scheme, for more detailed information one can refer to Ref.[47].

where

and

Dropping the term $\tau$ in (2.2), it is the standard second-order central difference scheme. In order to get fourth-order scheme, we have to handle the high-order derivative terms $\frac{\partial^3 u }{\partial x^3}$ and $\frac{\partial^4 u}{\partial x^4}$. According to the governing equation (1.1), there is

Taking the first and second derivatives to (2.3) respectively, we have

and

where

Substituting (2.4) and (2.5) into (2.2) and approximating the first- and second-derivatives with the central difference, we can get the fourth-order compact scheme:

where

2.1.   Interface fitted with mesh

When the interface is in contact with one of the mesh nodes, it is referred to as the body-fitted case. Without loss of generality, we assume that the interface is located the mesh node $k$, as shown in Figure 1.

In this case, the irregular nodes are $k-1$ and $k+1$, and the compact stencil of these two irregular nodes are shown in Figure {2}. Then, the high order compact scheme at these two irregular nodes can be given as

and

Apparently, there are two auxiliary qualities $u^{-}$ and $u^{+}$ in the schemes. To apply the above scheme, it is necessary to approximate the $u^{-}$ and $u^{+}$ with adequate accuracy. We prepare to use the interface jump connecting conditions and establish the linear systems with $u^{+}$ and $u^{-}$ as variables. The difficulty lies in the high-order discretization of the first-order derivatives $u^{+}_{x}$ and $u^{-}_{x}$ on both sides of the interface.

Expand the $u_{k+1}$ and $u_{k-1}$ at the interface $\alpha$ by using the Taylor series, it reads

and

Rewritting the above two formulas, we can get the representations of $\frac{\partial u}{\partial x}\Big{|}^{+}$ and $\frac{\partial u}{\partial x}\Big{|}^{-}$ as follows

According to the governing equation, we have

Keeping the first two terms on the right-hand side of (2.12) and substituting (2.14) into (2.12), the second-order accurate approximate scheme of $\frac{\partial u}{\partial x}\Big{|}^{+}$ can be given as

where

To obtain the higher-order approximate of $\frac{\partial u}{\partial x}\Big{|}^{+}$, it is necessary to handle the third- and fourth-order derivatives in (2.12). From (2.4) and (2.5), the expressions of $u^{+}_{xxx}$ and $u^{+}_{xxxx}$ can be given as

and

where

Keeping the first three terms on the right-hand side of (2.12), we establish the third-order accurate approximate of $\frac{\partial u}{\partial x}\Big{|}^{+}$ as follows:

where

Further, the fourth-order approximate of $\frac{\partial u}{\partial x}\Big{|}^{+}$ can be given as

where

Similarly, the second-, third- and fourth-order approximation of $\frac{\partial u}{\partial x}\Big{|}^{-}$ can be given as

where

2.2.   Interface unfitted with mesh

When the interface is located within one of the computational mesh, it appears as shown in Figure 3. The irregular nodes are $k$ and $k+1$, and the construction of the higher order scheme is more complicated. Owing to the interface arbitrarily cutting through the computational mesh, the computational mesh of these irregular nodes is nonuniform, as shown in Figure 4.

We assume the function $u(x)$ is smooth enough, expand the $u(x_{i+1}), u(x_{i-1})$ at point $x_i$ by Taylor series

and

From (2.23) and (2.24), we can obtain

and

Define the difference operators

Equations (2.25) and (2.26) can be rewritten as

and

Substituting Eqs (2.29) and (2.30) into the governing equation (1.1), we can get the difference scheme for the elliptic interface equation

where

and

To obtain the higher-order accuracy discrete scheme, it is necessary to handle the third- and fourth-order derivatives of the term $\tau$. Substituting (2.4) and (2.5) into (2.31) and rearranging it, we have

where

For the irregular points $k$ and $k+1$, we use the interface $\alpha$ as one of the points in the compact stencil as shown in Figure 5. Thus, the 3-point compact stencil for nodes $k$ and $k+1$ are $k-1, k, \alpha^{-}$ and $\alpha^{+}, k+1, k+2$, respectively. From the above analysis, the difference operators for the first- and second-order derivatives for nodes $k$ and $k+1$ are respectively as follows

and

Substituting (2.34) and (2.35) into (2.33), we can get the higher-order compact scheme for the irregular point $k$:

where

Similarly, the higher-order compact scheme for the irregular point $k+1$ is as follows:

where

From the above analysis, we have constructed the higher-order compact scheme at regular points and irregular points. It should be noted that there are two auxiliary unknown quantities $u^{+}$ and $u^{-}$ in the scheme for the irregular points $k$ and $k+1$. Therefore, we have to find a way to deal with the interface connection conditions, so as to obtain a high-precision approximation for the auxiliary unknowns $u^{+}$ and $u^{-}$ on both sides of the interface.

3.   Approximation of the normal derivative on the interface

The first order derivative on both sides of the interface, $\frac{\partial u}{\partial x}\Big{|}^{+}$ and $\frac{\partial u}{\partial x}\Big{|}^{-}$, can respectively be given as

and

where $h_{l1} = \alpha - x_k$, $h_{r1} = x_{k+1} - \alpha$, $h_{l2} = h_{l1} + h$ and $h_{r2} = h_{r1} + h$, as shown in Figure 6.

Further, the second-order derivative on both sides of the interface, $\frac{\partial^2 u}{\partial x^2}\Big{|}^{+}$ and $\frac{\partial^2 u}{\partial x^2}\Big{|}^{-}$, can respectively be given as

and

Substituting (3.3) and (3.4) into (3.1) and (3.2) respectively, we can get

and

3.1.   Second-order scheme

Keep the first term in (3.6) and (3.5) and discard the other terms. We can get the approximation of the first order derivative on both sides of the interface with second-order accuracy:

and

where

3.2.   Third-order scheme

Keeping the first two terms in (3.6) and (3.5) and discarding the other terms, and according to the (2.16), we can get the approximation of the first derivative with third order accuracy

and

where

3.3.   Fourth order scheme

Following the same manner as in the above subsection, we can get the approximation of the first derivative with fourth-order accuracy

and

where

4.   Compute $u^{+}$ and $u^{-}$ on the interface

The implicit connecting condition on the interface is given as

For the interface fitted mesh, the approximation of the first-order derivative on both sides of the interface can respectively be written as

Discarding the terms $O(h^{m}_l)$ and $O(h^{m}_r)$ in (4.2) and (4.3), and substituting them into (4.1), the discrete format of the implicit connecting condition on the interface can be given as

For brevity, rearrange the above equation and establish linear equations with $u^{+}$ and $u^{-}$ as variables as follows:

where

Solving the above linear system, we can get the approximate of each of the unknowns $u^{+}$ and $u^{-}$ on both sides of the interface with $m$-th order accuracy:

where

For the interface-cut mesh, the approximation of the first-order derivative on both sides of the interface can be written as

In a similar manner as above, the approximate of each of the unknowns $u^{+}$ and $u^{-}$ on both sides of the interface with $m$-th-order accuracy is given as

where

and

5.   Discrete system

From the above sections, we establish the fourth-order compact finite-difference scheme for non-irregular mesh nodes:

for

In the interface-fitted mesh case, the higher-order compact schemes at the irregular mesh nodes $k-1$ and $k+1$ are given as (2.7) and (2.8), respectively. Eliminating the auxiliary qualities $u^{+}$ and $u^{-}$ by substituting the formulas (4.4) and (4.5) into (2.7) and (2.8). The schemes for the irregular mesh nodes are given as

and

where

Similarly, in the non-body-fitted case, the higher-order compact schemes at the irregular mesh nodes $k$ and $k+1$ are respectively given as (2.36) and (2.37). Substitute the formulas (4.8) and (4.9) into (2.36) and (2.37) to eliminate the auxiliary qualities $u^{+}$ and $u^{-}$. The higher-order compact schemes at the irregular mesh nodes $k$ and $k+1$ are respectively given as

and

where

Obviously, the resulting discrete linear equations have a tri-diagonal form for the body-fitted case and block-diagonal form for the non-body-fitted case, which can be efficiently solved by using existing numerical methods, such as the forward and backward sweep method, BiCGstab method, etc.

6.   Numerical experiments

In this section, we use several numerical experiments to demonstrate the performance of the discrete schemes.

Example 1. Consider the computational domain $\Omega = [0, 1]$, and the solution is separated into two parts by the interface at $x = \alpha$, where $\alpha = 0.5$ and $\alpha = 0.52323$, for the interface-fitted mesh and interface-cut mesh cases, respectively. The analytical solution of this problem is given by

The diffusion coefficient is defined as follows

Conservation of the flux on the interface satisfies

The coefficient $\lambda$ is given as

Tables 1 and 2 compare the $L_{2}$ and $L_{\infty}$ errors in second-, third- and fourth-order formats on fitted and non-fitted mesh, respectively. The tables show that the numerical results of the three methods achieve theoretical accuracy on both fitted and non-fitted mesh.

Due to the strong discontinuity of physical quantities at the interface, it is crucial for the numerical scheme to be stable and robust. Tables 3 and 4 compare the errors of $u^{+}$ and $u^{-}$ for different methods on fitted and non-fitted mesh, respectively. It can be seen from the tables that the physical quantities on both sides of the interface differ in magnitude by a factor of 100, but the numerical scheme proposed can capture the discontinuity of physical quantities on both sides of the interface with a high degree of accuracy.

Figure 7 compares the exact and numerical solutions on fitted and non-fitted mesh for Problem 1 at $k = 100$ with a number of intervals of $32$. It is shown that the numerical solutions are well matched with the exact solutions, in spite of the fact that the amount of physical jumps on both sides of the interface are much larger. Figures 8 and 9 errors by the fourth-order, second-order and third-order scheme on fitted and non-fitted mesh, respectively. It can be seen that the errors are decrease as the number of mesh points increases.

Example 2. Consider the elliptic interface problem with a variable diffusion coefficient, and the solution is separated into two parts by the interface at $x = \alpha$, where $\alpha = 0.5$ and $\alpha = 0.53232$, for the interface-fitted mesh and interface-cut mesh cases, respectively. The analytical solution of this problem is given by

The diffusion coefficient is defined as follows

Conservation of the flux on the interface satisfies

The coefficient $\lambda$ is given as

Tables 5 and 6 compare the $L_{2}$ and $L_{\infty}$ errors in second-, third- and fourth-order formats on fitted and non-fitted mesh, respectively. It can be seen that the numerical format is able to achieve theoretical accuracy on fitted mesh. The numerical accuracy on the non-fitted mesh is slightly lower than the theoretical accuracy, probably due to the unbalanced grid step on both sides of the interface, which leads to the instability of the format.

Figure 10 compares the exact and numerical solutions on fitted and non-fitted mesh for Problem 1 at $k = 100$ with a number of intervals of $41$. It is shown that the numerical solutions are well matched with the exact solutions, in spite of the fact that the amount of physical jumps on both sides of the interface are much larger. Figure 11 compares errors by the fourth-order and third-order schemes on fitted and non-fitted mesh, respectively. As seen in the table, the errors in the fourth-order scheme are significantly lower than those in the third-order scheme.

Table 7 compares the length ratio of the interface-cut mesh interval. The table shows that as the number of grids changes, the spacing ratio changes significantly and affects the accuracy of the numerical scheme.

Table 8 compares the condition numbers of the linear system matrix $A$ for the fitted and non-fitted meshes. It can be seen that the condition number of the matrix does not differ much for the same grid number, so the proposed numerical scheme is stable.

Example 3. Consider the elliptic interface problem with a variable diffusion coefficient, and the solution is separated into two parts by the interface at $x = \alpha$, where $\alpha = 0.5$ and $\alpha = 0.53232$, for the interface-fitted mesh and interface-cut mesh cases, respectively. The analytical solution of this problem is given by

The diffusion coefficient is defined as follows

Conservation of the flux on the interface satisfies

The coefficient $\lambda$ is given as

Tables 9 and 10 compare the $L_{2}$ and $L_{\infty}$ errors in second-, third- and fourth-order formats on fitted and non-fitted mesh, respectively. It is shows that the numerical format is able to achieve theoretical accuracy on fitted mesh.

Figure 12 compares the exact and numerical solutions on fitted and non-fitted mesh for Problem 1 at $k = 100$ with a number of intervals of $41$. It can be seen that the error in matching the numerical solution to the exact solution is essentially small despite the large physical jumps on both sides of the interface. Figure 13 compares errors by the fourth-order and third-order schemes on fitted and non-fitted mesh, respectively. As can be seen in the table, the numerical accuracy of the fourth-order scheme is better than that of the third-order scheme.

7.   Conclusions

The 1D elliptic interface problem with imperfect contact is characterized by the fact that the jump quantity of the solution is unknown and is related to the flux across the interface. In this paper, a class of higher-order finite-difference schemes is constructed for interface fitted and non-fitted meshes, respectively. The second-, third-, and fourth-order approximations of the jump conditions are provided by using the connected jump conditions and their higher-order derivatives. Some numerical experiments were carried out to illustrate the accuracy and stability of the present method. The numerical results show that the proposed scheme is able to capture the solution of the discontinuous case of the interface body with theoretical accuracy. The theoretical accuracy has been achieved for the elliptic interface problem with implicit interface connection conditions.

Acknowledgments

This work was partly supported by the National Natural Science Foundation (12161067, 12261067, 12001015, 62201298, 51961031), the National Natural Science Foundation of China Youth Fund Project (11801287), the Inner Mongolia Autonomous Region Youth Science and Technology Talents Support Program (NJYT20B15), the Inner Mongolia Scientific Fund Project (2020MS06010, 2021LHMS01006, 2022MS01008), and Innovation Fund Project of Inner Mongolia University of Science and Technology-Excellent Youth Science Fund Project (2019YQL02).

Conflict of interest

The authors declare that the publication of this paper does coincide with any confilict of interest.