Introduction

Singularly perturbed problems frequently occur in fluid mechanics, combustion theory, plasma physics, and indifferent day-to-day physical phenomena. Those problems that show a rapid change in its solutions that inherently contains very sharp layers, (see the books in [1, 2]). As those books explain, in singularly perturbed problems, a very small positive parameter called singular perturbation parameter is multiplied to the highest order derivative term and as this parameter goes smaller and smaller, layer occurs. Then the solution shows a much-unexpected change in a very small portion of the domain. In such a small portion, it becomes challenging for numerical methods to capture the solution accurately.

Singularly perturbed parabolic problems are also kinds of singularly perturbed problems that arise in various branches of science and engineering. The well-known examples are the Navier–Stokes equation with large Reynolds number in fluid dynamics, the convective heat transport problems with large Péclet number, and so on. Numerical treatment of the singularly perturbed problems is difficult because of the presence of boundary and/or interior layers in its solution. In particular, classical finite difference or finite element methods fail to yield satisfactory numerical results on uniform meshes and to obtain stability concerning the perturbation parameter, [3,4,5,6,7].

The fitted numerical methods for solving the singularly perturbed problems are widely classified into the fitted operator and fitted mesh methods. In fitted operator methods, exponential fitting parameters will be used to control the rapid growth or decay of the numerical solution in layer regions. Whereas, fitted mesh methods use nonuniform meshes, which will be fine in layer regions and coarse outside the layer regions, [6,7,8,9,10,11]. Thus, many fitted numerical methods developed for different types of singularly perturbed parabolic problems. These different types of problems are occurred depending on whether or not the existence of the convection term, the number of parameters, layer type (boundary and/or interior layers), and the dimension of the problems, linearity, and so on. For the detailed types of the singularly perturbed family of parabolic problems and the developed methods, one can refer to the literature in [12,13,14,15,16,17,18,19] in addition to the formerly listed references.

In the past few decades, few numerical schemes are presented for solving singularly perturbed parabolic convection–diffusion problem types [4,5,6,7,8,9,10,11]. These referred articles may help us just to get prior knowledge about the nature of the solution of these families of problems and where and why the existing methods in problematic to work. Further, it is a recent and active research area in engineering and applied science. Though many classical numerical methods such as finite difference methods, finite element methods, and finite volume methods have been developed so far, most of them fail to give a more accurate solution. This difficulty is due to the presence of perturbation parameter that causes the existence of a boundary layer where the solutions vary rapidly and behave smoothly away from the layer. Owing to the classical numerical methods cannot give a more accurate solution for singularly perturbed parabolic problems, [1, 2], researchers provide attention to formulate methods that may give a more accurate solution.

Recently, researchers in [9, 10], were presented different types of parameter-uniform numerical methods to solve singularly perturbed parabolic problems that exhibiting twin boundary layers. They study the asymptotic behavior of the solution and its partial derivatives. The problem is discretized using the implicit Euler method for time discretization on a uniform and nonuniform mesh with a hybrid scheme for spatial discretization on a generalized Shishkin mesh. The schemes were shown to be uniformly convergent of order one in the time direction and order two in the spatial direction up to a logarithmic factor. Numerical experiments are conducted to validate the theoretical results. Comparison is done with the upwind scheme on a uniform mesh as well as on the standard Shishkin mesh to demonstrate the higher-order accuracy of the proposed scheme on a generalized Shishkin mesh. Also, more recently, researchers in [20,21,22], were provided several novel parameter-uniform higher order numerical methods to solve different types singularly perturbed problems.

Hence, in the past few decades, various numerical schemes are proposed to solve families of these problems, but the obtained results yet not satisfactory for the problem under consideration. Thus, it is necessary to formulate and analyze layer resolving fitted mesh numerical method to produce accurate numerical solutions for the mentioned problem. Therefore, the main aim of this paper is to answer the questions raised related to the accuracy of the solution, and stability and consistency of the method for the singularly perturbed parabolic problem with twin boundary layers.

Statement of the problem

Singularly perturbed parabolic problems are maybe broadly categorized into problems of reaction–diffusion and convection–diffusion types. The convection–diffusion type has also its different types depending on the kind of layers (boundary and/or interior layers). Henceforth, singularly perturbed parabolic convection–diffusion problems are divided into problems exhibiting right or left boundary layer, interior layer, boundary and interior layers. In this work, we proposed a layer resolving fitted numerical scheme for solving the singularly perturbed parabolic of the convection–diffusion type that exhibits twin boundary layers of the problem:

$$\varepsilon \frac{{\partial^{2} u}}{{\partial x^{2} }} + a(x)\frac{\partial u}{{\partial x}} - b(x)u(x,t) - \frac{\partial u}{{\partial t}} = f(x,t),\,\,\,\,\forall (x,t) \in D,$$
(1)

This is subject to the initial and boundary conditions:

$$\begin{gathered} u(x,0) = u_{0} (x),\,\,\,\,\forall x \in [x_{l} ,x_{r} ], \hfill \\ u(x_{l} ,t) = q_{l} (t),\,\,\,\,\forall t \in [0,1], \hfill \\ u(x_{r} ,t) = q_{r} (t),\,\,\,\,\forall t \in [0,1], \hfill \\ \end{gathered}$$
(2)

Here the solution domain is \(D: = (x_{l} ,\,x_{r} ) \times (0,\,1]\) and \(\varepsilon ,\,\,\,0 < \varepsilon < < 1,\) is the perturbation parameter. Sufficient regularity conditions are imposed on the data’s in Eqs. 1 and 2 that guarantee the smoothness of the solution on the set \(\overline{D}\). Also, assume that Eq. 1 has only one turning point. That is, the coefficient of convection term \(a(x)\) vanishes exactly at one value of \(d = \frac{{x_{l} + x_{r} }}{2}\). For the uniqueness of the solution to Eq. 1, assume that the functions are sufficiently smooth and satisfy the conditions, [10]:

$$\left\{ \begin{gathered} a(d) = 0, \hfill \\ a^{\prime}(d) < 0, \hfill \\ a(x) < 0,\,\,\,\,\forall x \in [x_{l} ,\,d), \hfill \\ a(x) > 0,\,\,\,\,\forall x \in (d,\,x_{r} ], \hfill \\ b(x) \ge \beta > 0,\,\,\forall x \in [x_{l} ,\,x_{r} \,], \hfill \\ \frac{b(d)}{{a^{\prime}(d)}}\,\, < \,\,0. \hfill \\ \end{gathered} \right.$$
(3)

The conditions provided in Eq. 3 are used to indicate the layer regions located both at the ends of the spatial domain. Due to the classical numerical methods cannot give accurate solution for problem under consideration, researchers provide attention to formulate methods that may give a more accurate solution. Hence, the main objective of this paper to present a type of fitted mesh numerical scheme to produce a more accurate numerical solution for singularly perturbed parabolic convection–diffusion problem exhibiting twin boundary layers. Moreover, the detailed lemmas with its proofs of the existence and uniqueness of the solution for the problems defined in Eqs. 1–3 is provided in [10]. Furthermore, to get the existence uniqueness, compatibility and methods, researchers recommended to track the methodology in the works, [31,32,33,34,35,36,37,38].

Formulation of the numerical scheme

To formulate the scheme, we first discretize the temporal variable on uniform mesh and then discretize the spatial one on piecewise uniform Shishkin mesh type. The partition of time interval \([0,T]\) with uniform step size k is given by

$$t_{n} = nk,\,\,\,0 \le n \le N,\,\,\,\,k = \frac{T}{N}.$$
(4)

Now, using the back Euler approach, we obtain a system of linear differential equations:

$$\left\{ \begin{gathered} \varepsilon \frac{{\partial^{2} U(x,t_{n} )}}{{\partial x^{2} }} + a(x)\frac{{\partial U(x,t_{n} )}}{\partial x} - [b(x) + \frac{1}{k}]U(x,t_{n} ) = H(x,t_{n} ), \hfill \\ \,\,\,\,\,U(x,0) = u_{0} (x),\,\,\,\,\forall x \in [x_{l} ,x_{r} ], \hfill \\ \,\,\,\,\,U(x_{l} ,t_{n} ) = q_{l} (t_{n} ),\,\,\,\,\forall \,0 \le n \le N, \hfill \\ \,\,\,\,\,U(x_{r} ,t_{n} ) = q_{r} (t_{n} ),\,\,\,\,\forall \,0 \le n \le N, \hfill \\ \end{gathered} \right.$$
(5)

Here \(H(x,t_{n} ) = f(x,t_{n} ) - \frac{1}{k}U(x,t_{n - 1} )\). This gives the semi-discretize approximation \(u(x,t_{n} )\) to the exact solution \(u(x,t)\) of Eq. 1 at the time levels \(t_{n} = nk\).

Remarks

  1. I.

    The estimate of local error in the temporal direction is given by.

    $$\left| {E_{n} } \right| \le Ck^{2}$$
  2. II.

    The estimate of the global error in the direction of time is given by

    $$\left\| {E_{n} } \right\| \le Ck,\,\,\forall n$$

Here C is a positive constant free from perturbation parameter and mesh size k.

Consider the solution to Eq. 5 has large gradients in a narrow region near \(x = x_{l}\) and \(x = x_{r}\), then the mesh in this region will be fine and coarse everywhere else. Let \(M\) be a positive integer such that \(M \ge 8\). With this in mind, the transition positive parameter \(\tau\) is chosen to be

$$\tau = \min \left\{ {\frac{1}{4},\,\,\,\varepsilon \ln \left( M \right)} \right\}.$$
(6)

Assume that \(\varepsilon \le \frac{1}{M}\), and considering the sub-intervals \([x_{l} ,\,x_{l} + \tau ]\), \([x_{l} + \tau ,\,\,x_{r} - \tau ]\) and \([x_{r} - \tau ,\,\,x_{r} ]\) of the interval \([x_{l} ,\,x_{r} ]\) are subdivided uniformly to contain \(\frac{M}{4},\,\frac{M}{2}\) and \(\frac{M}{4}\) mesh elements. The partition of interval \([x_{l} ,\,x_{r} ]\) is defined by:

$$\left\{ \begin{gathered} x_{0} = x_{l} . \hfill \\ x_{m} = x_{l} + mh_{m} ,\,\,\,\,\,\,m = 1,2,\,\,...\,\,,M - 1, \hfill \\ x_{M} = x_{r} . \hfill \\ \end{gathered} \right.$$
(7)

The mesh spacing \(h_{m} = x_{m} - x_{m - 1}\) is given by:

$$h_{m} = \left\{ {\begin{array}{*{20}c} {\frac{{4(x_{l} + \tau )}}{M},\,\,\,\,{\text{for}}\,\,m = 1,2,...,\frac{M}{4},\,\frac{3M}{4} + 1,...,M,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ \begin{gathered} \frac{{2(x_{r} - x_{l} - 2\tau )}}{M},\,\,\,{\text{for}}\,\,\,m = \,\,\,\frac{M}{4} + 1,\frac{M}{4} + 2,\,\,...\,,\,\,\,\frac{3M}{4},\,\, \hfill \\ \frac{{4(x_{r} - \tau )}}{M},\,\,\,\,{\text{for}}\,\,\,m = \,\frac{3M}{4} + 1,...,M. \hfill \\ \end{gathered} \\ \end{array} } \right.$$
(8)

Representing this mesh by \(D_{M}^{N}\) and for the rest of the paper, any function \(F(x,t)\) adopt the notation \(F(x_{m} ,t_{n} ) = F_{m}^{n}\). Then, the discretize form of the problem in Eq. 5 on \(D_{M}^{N}\) as:

$$\left\{ \begin{gathered} \varepsilon \delta_{x}^{2} U_{m}^{n} + a_{m} \delta_{x}^{ + } U_{m}^{n} - [b_{m} + \frac{1}{k}]U_{m}^{n} = H_{m}^{n} ,\,\,\,\,m = 1,\,2,\,...\,\,\frac{M}{2},\,\,\,\,\forall n, \hfill \\ \varepsilon \delta_{x}^{2} U_{m}^{n} + a_{m} \delta_{x}^{{^{ - } }} U_{m}^{n} - [b_{m} + \frac{1}{k}]U_{m}^{n} = H_{m}^{n} ,\,\,\,\,m = \frac{M}{2} + 1,\,\,....\,\,M - 1,\,\,\,\,\forall n, \hfill \\ \,\,\,\,\,U_{m}^{0} = u_{0} (x_{m} ),\,\,\,\,\forall x_{m} \in [x_{l} ,x_{r} ], \hfill \\ \,\,\,\,\,U_{0}^{n} = q_{l} (t_{n} ),\,\,\,\,\forall \,0 \le n \le N, \hfill \\ \,\,\,\,\,U_{M}^{n} = q_{r} (t_{n} ),\,\,\,\,\forall \,0 \le n \le N, \hfill \\ H_{m}^{n} = f_{m}^{n} - \frac{1}{k}u_{m}^{n - 1} ,\,\,\,\,\forall \,\,m,n, \hfill \\ \end{gathered} \right.$$
(9)

Here \(\delta_{x}^{2} U_{m}^{n} = \frac{2}{{h_{m} + h_{m + 1} }}\left( {\delta_{x}^{ + } U_{m}^{n} - \delta_{x}^{{^{ - } }} U_{m}^{n} } \right), \ldots \delta_{x}^{ + } U_{m}^{n} = \frac{{U_{m + 1}^{n} - U_{m}^{n} }}{{h_{m + 1} }}\)

To make more clearly, the scheme in Eq. 9 can be re-written in the form:

$$E_{m} U_{m - 1}^{n} + F_{m} U_{m}^{n} + G_{m} U_{m + 1}^{n} = H_{m}^{n} ,\,\,\,\forall n,$$
(10)

Here for \(m = 1,\,2,\,...\,\,\frac{M}{2},\,\,\,\)

$$E_{m} = \frac{2\varepsilon }{{h_{m} (h_{m} + h_{m + 1} )}},\,\,\,\,\,F_{m} = \frac{ - 2\varepsilon }{{h_{m} h_{m + 1} }} - \frac{{a_{m} }}{{h_{m + 1} }} - b_{m} - \frac{1}{k},\,\,\,\,\,G_{m} = \frac{2\varepsilon }{{h_{m + 1} (h_{m} + h_{m + 1} )}} + \frac{{a_{m} }}{{h_{m + 1} }},$$

In addition, for \(m = \frac{M}{2} + 1,\,\,...\,\,M - 1,\,\,\)

$$E_{m} = \frac{2\varepsilon }{{h_{m} (h_{m} + h_{m + 1} )}} - \frac{{a_{m} }}{{h_{m} }},\,\,\,\,\,\,\,F_{m} = \frac{ - 2\varepsilon }{{h_{m} h_{m + 1} }} - \frac{{a_{m} }}{{h_{m} }} - b_{m} - \frac{1}{k},\,\,\,\,\,G_{m} = \frac{2\varepsilon }{{h_{m + 1} (h_{m} + h_{m + 1} )}}.$$

Stability and consistency of the scheme

To solve these recurrence relations in Eq. 10, we apply the Thomas algorithm regards to the space direction. Further, the conditions for the discrete invariant imbedding algorithm to be stable, if and only if:

$$\left| {F_{m} } \right| \ge \,\left| {E_{m} } \right| + \,\left| {G_{m} } \right|,$$
(11)

This inequality is strictly satisfied, since \(b_{m} + \frac{1}{k} > \,\,0,\,\,\,\forall m\). Hence, the Thomas Algorithm is stable for the described numerical scheme.

The truncation error \(T\) between the exact solution \(u(x_{m} ,\,t_{n} )\), and the approximate solution \(U_{m}^{n}\) is given by:

$$\begin{gathered} T = (\varepsilon \frac{{\partial^{2} }}{{\partial x^{2} }} + a(x_{m} )\frac{\partial }{\partial x} - b(x_{m} ) - \frac{\partial }{\partial t})u(x_{m} ,\,\,t_{n} ) - \left( {\varepsilon \delta_{x}^{2} U_{m}^{n} } \right.\,\, + a_{m} \delta_{x}^{*} U_{m}^{n} - b_{m} U_{m}^{n + 1} - \frac{{U_{m}^{n} - U_{m}^{n - 1} }}{k}\left. {} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\ \end{gathered}$$
(12)

Here \(\delta_{x}^{*} U_{m}^{n} = \delta_{x}^{ + } U_{m}^{n} ,\,\,\,m = 1,2,...,\,\frac{M}{2}\), and \(\delta_{x}^{*} U_{m}^{n} = \delta_{x}^{{^{ - } }} U_{m}^{n} ,\,\,\,m = \frac{M}{2} + 1,\,\,...\,\,M - 1.\)

Assume that the higher order derivatives of \(U(x,t)\) exists with respect to the two independent variables and using Taylor’s series expansions, we have:

$$\delta_{x}^{ - } U_{m}^{n} \approx \frac{{\partial U_{m}^{n} }}{\partial x} - \frac{{h_{m} }}{2!}\frac{{\partial^{2} U_{m}^{n} }}{{\partial x^{2} }} + \frac{{h_{m}^{2} }}{3!}\frac{{\partial^{3} U_{m}^{n} }}{{\partial x^{3} }} + ...$$
(13)
$$\delta_{x}^{ + } U_{m}^{n} \approx \frac{{\partial U_{m}^{n} }}{\partial x} + \frac{{h_{m + 1} }}{2!}\frac{{\partial^{2} U_{m}^{n} }}{{\partial x^{2} }} + \frac{{h_{m + 1}^{2} }}{3!}\frac{{\partial^{3} U_{m}^{n} }}{{\partial x^{3} }} + ...$$
(14)
$$\delta_{x}^{2} U_{m}^{n} = \frac{2}{{h_{m} + h_{m + 1} }}\left( {\delta_{x}^{ + } U_{m}^{n} - \delta_{x}^{ - } U_{m}^{n} } \right)\,\,\,\,\,\, \approx \,\,\,\frac{{\partial^{2} U_{m}^{n} }}{{\partial x^{2} }} - \frac{{h_{m + 1} - h_{m} }}{3}\,\frac{{\partial^{3} U_{m}^{n} }}{{\partial x^{3} }}\,\, + ...$$
(15)
$$\frac{{U_{m}^{n + 1} - U_{m}^{n} }}{k} \approx \frac{{\partial U_{m}^{n} }}{\partial t} + \frac{k}{2}\frac{{\partial^{2} U_{m}^{n} }}{{\partial t^{2} }} + \frac{{k^{2} }}{3!}\frac{{\partial^{3} U_{m}^{n} }}{{\partial t^{3} }} + O(k^{3} )$$
(16)

Substituting Eqs. 13–16 into Eq. 12 yields the estimated truncation error:

$$\begin{gathered} T = \left\{ \begin{gathered} \frac{{a_{m} h_{m} }}{2}\frac{{\partial^{2} U_{m}^{n} }}{{\partial x^{2} }} - \frac{k}{2}\frac{{\partial^{2} U_{m}^{n} }}{{\partial t^{2} }} + ...,\,\,\,\,m = 1,2,...,\frac{M}{2}, \hfill \\ \frac{{a_{m} h_{m + 1} }}{2}\frac{{\partial^{2} U_{m}^{n} }}{{\partial x^{2} }} - \frac{k}{2}\frac{{\partial^{2} U_{m}^{n} }}{{\partial t^{2} }} + ...,\,\,m = \frac{M}{2} + 1,...,M - 1. \hfill \\ \end{gathered} \right. \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\ \end{gathered}$$
(17)

From the considered piecewise discretization or from Eq. 8 of the solution domain, assume that.

\(h_{m} = \frac{{4(x_{l} + \tau )}}{M}\) and the value of chosen transition parameter is.

$$\tau = \varepsilon \;\ln \,\left( M \right)\; \le \;\frac{4}{M}\,\ln \left( M \right)$$

Thus, in the neighborhoods between the inner and outer layer region, we have:

$$h_{m} = \frac{{4(x_{l} + \tau )}}{M} = \frac{{4(x_{l} + \varepsilon \ln \left( M \right))}}{M}\, \le \frac{{4x_{l} }}{M} + \frac{16\ln \left( M \right)}{{M^{2} }}\,\,\, \le \,\,M^{ - 1} \ln \left( M \right)$$
(19)

Thus, from Eqs. 17 and 19 the norm of truncation error for the formulated scheme is:

$$\left\| T \right\| \le C\left( {M^{ - 1} \ln \left( M \right) + k} \right),$$
(20)

Here the bounded error \(C = \frac{1}{2}\left( {\left\| {a_{m} \frac{{\partial^{2} U_{m}^{n} }}{{\partial x^{2} }}} \right\|_{\infty } + \left\| {\frac{{\partial^{2} U_{m}^{n} }}{{\partial t^{2} }}} \right\|_{\infty } } \right)\) is a constant.

Therefore, the described scheme is almost first-order convergent. From its definition, truncation errors measure how well a finite difference scheme approximates the differential equation. Thus, the method is almost first-order accurate. A finite difference method is consistent if the limit of truncation error is equal to zero as the mesh size goes to zero. Thus, using this consistency and stability criteria provided in Eq. 11, the proposed method is convergent by Lax’s equivalence theorem.

Richardson extrapolation

Richardson extrapolation is that whenever the leading term in the error for an approximation scheme is known. This procedure is a convergence acceleration technique that consists of considering a linear combination of two computed approximations of a solution. The rigorous proof on optimal uniform analysis and also for extrapolation approaches, one cans the mile-stone works in [23,24,25,26,27,28,29,30]. Since, the described numerical scheme is almost second-order convergent as verified in Eq. 20, we have:

$$\left| {u(x_{m} ,t_{n} ) - U_{m}^{n} } \right| \le C\left( {M^{ - 1} \ln (M) + k} \right),$$
(21)

Here \(u(x_{m} ,t_{n} )\) and \(U_{m}^{n}\) are exact and approximate solutions, C is a constant independent of mesh sizes \(h_{m} ,\,k\) and perturbation parameter \(\varepsilon\). Let us be the mesh obtained by bisecting each mesh interval and considering Eq. 21 works for any \(h_{m} ,\,k \ne 0\), which implies:

$$u(x_{m} ,t_{n} ) - U_{m}^{n} \approx C\left( {h_{m} \ln (M) + k} \right) + R^{M,N} .$$
(22)

So as to, it works for any \(\frac{{h_{m} }}{2},\frac{k}{2} \ne 0\) yields:

$$u(x_{m} ,t_{n} ) - U_{2m}^{2n} \approx C\left( {\frac{{h_{m} }}{2}\ln (2M) + \frac{k}{2}} \right) + R^{2M,\,\,2N} .$$
(23)

Here the remainders, \(R^{M,N}\) and \(R^{2M,\,2N}\) are \(O(h_{m}^{4} \ln (M) + k^{4} )\).

Reducing the constant C from Eqs. 22 and 23 leads to

$$u(x_{m} ,t_{n} ) - (2U_{2m}^{2n} - U_{m}^{n} ) \cong O(h_{m}^{2} \ln (M) + k^{2} ),$$

Thus suggests that the following value is also an approximation of \(\,u(x_{m} ,t_{n} )\).

$$\left( {U_{m}^{n} } \right)^{ext} = 2U_{2m}^{2n} - U_{m}^{n}$$
(24)

Using this approximation to evaluate the truncation error, we obtain:

$$\left| {u(x_{m} ,\,t_{n} ) - \left( {U_{m}^{n} } \right)^{ext} } \right| \le C(h_{m}^{2} \ln (M) + k^{2} )$$
(25)

This is the Richardson extrapolation technique to accelerate the almost first-order convergent to almost second-order convergent method.

Numerical illustrations

In this section, experimental illustrations are conducted on two sample examples to validate the efficiency and applicability of the presented method. In these considered examples a turning point happened at \(x = 0.5\) and \(x = 0\) for Examples 1 and 2, respectively. Further, depending on the sign of the coefficient of convection term regards to the left or right side of a turning point, the exhibits twin boundary layers. Ever since the exact solutions of the considered examples are not known, the double mesh principle is used to estimate the maximum absolute error as:

$$E_{M}^{N} = \mathop {\max }\limits_{\begin{subarray}{l} 0 \le m \le M\, \\ 0 \le n \le N \end{subarray} } \left| {U_{m}^{n} - U_{2m}^{2n} } \right|,$$

Here \(U_{m}^{n}\) and \(U_{2m}^{2n}\) are approximate solutions. The numerical rates of convergence evaluated by:

$$R_{M}^{N} = \frac{{\log (E_{M}^{N} ) - \log (E_{2M}^{2N} )}}{\log (2)}.$$

Example 1: Consider the singularly perturbed parabolic problem:

$$\left\{ \begin{gathered} \varepsilon \frac{{\partial^{2} u(x,t)}}{{\partial x^{2} }} - 2(2x - 1)\frac{\partial u(x,t)}{{\partial x}} - 4u(x,t) - \frac{\partial u(x,t)}{{\partial t}} = 0,\,\,\,\,\forall (x,t) \in (0,\,1) \times (0,\,1], \hfill \\ \,\,\,u(x,0) = 1,\,\,\,\,x \in [0,\,\,1], \hfill \\ \,\,\,u(0,t) = 1,\,\,\,\,\,t \in [0,\,\,1], \hfill \\ \,\,\,u(1,t) = 1,\,\,\,\,\,\,t \in [0,\,\,1]. \hfill \\ \end{gathered} \right.$$

Example 2: Consider the singularly perturbed parabolic problem:

$$\left\{ \begin{gathered} \varepsilon \frac{{\partial^{2} u(x,t)}}{{\partial x^{2} }} - x\frac{\partial u(x,t)}{{\partial x}} - \frac{\partial u(x,t)}{{\partial t}} - u(x,t) = 1,\,\,\,\,\forall (x,t) \in ( - 1,\,1) \times (0,\,1], \hfill \\ \,\,\,u(x,0) = 1,\,\,\,\,x \in [ - 1,\,\,1], \hfill \\ \,\,\,u( - 1,t) = 1,\,\,\,t \in [0,\,\,1], \hfill \\ \,\,\,u(1,t) = 1,\,\,\,\,\,\,\,t \in [0,\,\,1]. \hfill \\ \end{gathered} \right.$$

The maximum absolute error and the order of convergence are computed and provided in Tables, corresponding to different values of perturbation and mesh parameters. Fig. 1, demonstrates that the proposed method preserves the layer behaviour of the problem and matches its parabolic shape for the two examples under consideration.

Fig. 1
figure 1

Surface plot of numerical solutions when \(\varepsilon = 2^{ - 10} ,\,\,N = M = 32\) for Examples 1 and 2

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Discussions and conclusion

The present method is fitted mesh numerical method based on type of Shishkin meshes for solving singularly perturbed turning point parabolic problems with twin boundary layers. We have recognized the stability and consistency of the formulated scheme to guarantee the convergence of the method. Hence, the convergence analysis established that confirmed using numerical results Tables, which is almost first order convergent and accelerated to almost second-order convergent. Basically, results in Tables shows that the proposed method is fundamentally first order convergent and the error has monotonically decreasing behavior with increasing number of mesh intervals N and M, which approve convergence of proposed scheme. Comparison of numerical results in Table 1 shows that, the present scheme gives more accurate results than the scheme given in results in [10]. Figure 1, indicates the properties and interpretation of numerical solution to support the theoretical descriptions contain about twin boundary layers (See Tables 2, 3 and 4).

Table 1 Comparison of maximum absolute errors for Example 1
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Table 2 Computed maximum absolute errors for Example 1 regards to after and before applying Richardson extrapolation
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Table 3 Computed rate of convergence for Example 1 regards to after and before applying Richardson extrapolation
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Table 4 Maximum absolute errors for Example 2
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Generally, the present method is almost order one and accelerated to order two convergent in both the spatial and temporal variables up to a logarithmic factor, for solving singularly perturbed turning point parabolic problems with twin boundary layers. Further, the method is stable, convergent and gives more accurate solution than some existing methods in literature.