Edited by: Poom Kumam, King Mongkut's University of Technology Thonburi, Thailand
Reviewed by: Abdullahi Yusuf, Federal University Dutse, Nigeria; Kazuharu Bamba, Fukushima University, Japan
This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics
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In this paper Extrapolated Stabilized Explicit RungeKutta methods (ESERK) are proposed to solve nonlinear partial differential equations (PDEs) in right triangles. These algorithms evaluate more times the function than a standard explicit Runge–Kutta scheme (
Let us suppose that we have to solve a nonlinear PDE with dominating diffusion:
subject to traditional initial and Dirichlet boundary conditions:
and
These types of problems are very common in a large amount of areas such as atmospheric phenomena, biology, chemical reactions, combustion, financial mathematics, industrial engineering, laser modeling, malware propagation, medicine, mechanics, molecular dynamics, nuclear kinetics, etc., see [
A widelyused approach for solving these timedependent and multidimensional PDEs is to first discretize the space variables (with finite difference or spectral methods) to obtain a very large system of ODEs of the form
where
Hence, traditional explicit methods are usually very slow, due to absolute stability, it is necessary to use very small length steps, see [
Implicit schemes based on BDF and Runge–Kutta methods have much better stability properties. However, since the dimension of the ODE system is very high, then it is necessary to solve very large nonlinear systems at each iteration.
Other numerous techniques have also been proposed based on ETD schemes (but it is necessary to approximate operators including matrix exponentials), alternating direction implicit methods (they have limitations on the order of convergence) and explicitimplicit algorithms. However, in any case the number of operations is huge when the system dimension is high.
For those cases where it is known that the Jacobian eigenvalues of the function are all real negative or are very close to this semiaxis, there is another option: stabilized explicit Runge–Kutta methods (they are also called RungeKuttaChebyshev methods). This happens, for example, when diffusion dominates in the PDE, when the Laplacian is discretized using finite differences or some spectral techniques, then the associated matrix has this type of eigenvalues.
These types of algorithms are totally explicit, but they have regions of stability extended along the real negative axis. These schemes typically have order 2 or 4 [
These schemes have been traditionally considered in squares/rectangles or cubes. But this makes difficult to apply them in PDEs with complex geometries, which happens in most of the cases. Some different strategies have been proposed to apply them when the original domain is not a square nor a cube (see [
As far as we know, stabilized explicit Runge–Kutta methods have not been tested in triangles yet. For this reason, in this paper, we are analysing how ESERK methods can be employed to solve nonlinear PDEs in these types of regions and their convergence.
In this paper, a summary on ESERK4 methods is provided in section 2. The major advance of our contribution is given in section 3: it is explained how ESERK4 can be utilized for (1) when Ω is a right triangle. After some linear transformations and spatial discretizations ESERK4 is numerically stable and fourthorder convergence in time, and secondorder in space is obtained. This allows a new way to numerically approach parabolic nonlinear PDEs in complex domains in the plane, which can be easier decomposed in a sum of triangles and rectangles. Finally, some conclusions and future goals are outlined.
In [
If we consider
then 
We can construct Runge–Kutta schemes with 
Once firstorder SERK methods have been derived, they approximate the solution of the initial value problem (4), by performing
Finally
is a fourthorder approximation with
as its stability function. Additionally, we have that
The positive real solution of
is λ_{4} = 0.311688. Hence, whenever 
In [
Recently, we are working developing the parallel version of this code (see [
Complex geometric shapes are ubiquitous in our natural environment. In this paper, we are interested in numerically solving partial differential equations (PDEs) in such types of geometries, which are very common in problems related with human bodies, materials, or simply a complicated engine in classical engineering applications.
One very wellknown strategy, within a finite element context, is to build the necessary modifications in the vicinity of the boundary. Such an approach is studied in the composite finite element method (FEM). Those methods based on finite element are usually proposed only for linear PDEs. FEM is a numerical method for solving problems of engineering and mathematical physics (typical problems include structural analysis, heat transfer, fluid flow, mass transport, or electromagnetic potential, because these problems generally require numerically approximating the solution of linear partial differential equations). The finite element method allows the transformation of the problem in a system of algebraic equations. Unfortunately, it is more difficult to employ these techniques with nonlinear parabolic PDEs in several dimensions, although some results have been obtained to know when the resulting discrete Galerkin equations have a unique solution in [
On the other hand, Implicit–Explicit (IMEX) methods have been employed to solve a very stiff nonlinear system of ODEs coming from the spatial discretization of nonlinear parabolic PDEs that appeared in the modelization of an ischemic stroke in [
In what follows we will explain a new strategy to numerically solve the nonlinear parabolic PDE given by Equation (1) where Ω is any right triangle, and therefore any researcher can combine the theory (utilized with FEM) to spatially decomposed any complex geometry into triangles (since any acute triangle and obtuse triangle can be decomposed into two right triangles), and later employing the method described in this paper. Additionally, schemes proposed in this work are fourthorder ODE solvers (in time), and numerical spatial approximations will be secondorder (although fourthorder formulae can be explored except for the closest points to vertices).
Without loss of generality we can consider that our right triangle is
where the parameters
where
and it is easy to check that Det≠0 if and only the three points are not in a line (but we always have a triangle).
The main reason of decomposing our general region Ω∈
subject to (traditional) initial and Dirichlet boundary conditions. Therefore, let us first study Equation (12), together with
and
where ∂(
Now, let us define the spatial discretization of our continuous problem provided by Equation (12), the problem domain
since
With this semidiscretizations we will approximate
After the linear transformation given by Equations (9) and (10), our PDE given by Equation (1) may transform into one Equation where one term in
however, in
Spatial discretization in the right triangle
In this work, we are employing only secondorder approximations in space. In other works, for example [
and in the lower edge
However, in the triangle, again we can observe in
Now, we are ready to understand why we chose right triangles in the decomposition of complex regions. The main reason is, that simple calculations give us [after linear transformations given by Equations (9–11)]:
and therefore, after this linear transformation,
If we change
which is 0 if and only if the vectors
Thus, if the original triangle has a right angle at
Theorem: Let Equations (1)–(3) be the PDE to be solved, and Ω a right triangle with a right angle at
The associate matrix,
But simple calculations allow us to obtain that
where
0_{i, j} is the
Finally, it is wellknown that all the eigenvalues of any symmetric real matrix
Hence, equalizing real and imaginary parts, we have
and therefore
In this way we can conclude that
and, since 
Additionally, since σ, μ ≥ 0, the Gershgoring theorem guarantees for all the eigenvalues of
Therefore, whenever the nonlinear part does not modify this type of eigenvalues (real and negative) in the Jacobian function, a bound of the spectral radius of the Jacobian is 4(μ+σ), and we merely need to use an ESERK method with
Let us now study the numerical behavior of ESERK methods in a right triangle with one example. We will consider
where
Ω is the triangle with vertices (−1, 1), (−3, 2), and (0, 3) and initial and boundary conditions are taken such that
Hence, we first consider the linear transformation given by Equations (9) and (10), i.e.,
where
and initial and boundary conditions are taken such that
Now, it is possible to utilize secondorder approximations in space, as it was explained in the previous section. ESERK4 with
Analysis of the numerical convergence at points
2.264 
1.215 
1.595 

3.355 
6.364 
2.479 

4.430 
5.842 
3.109 

Spatial conv.  1.180  
3.449 
2.709 
4.073 

1.412 
8.117 
1.479 

1.649 
1.975 
4.536 

Spatial conv.  1.583 
Analysis of the numerical convergence at points
2.150 
1.228 
1.599 

3.235 
5.693 
2.132 

4.401 
5.842 
2.625 

Spatial conv.  1.303  
3.275 
3.657 
4.042 

1.327 
7.585 
1.788 

1.568 
1.975 
3.824 

Spatial conv.  1.701 
First of all, we calculated all the eigenvalues of the matrix
ESERK4 schemes are stable in [−
In both
If we take
Therefore, it is not so easy to observe 4−th order convergence in time and 2−nd in space. If we choose
Now, if we fix
Since, part of the error with
Analysis of the numerical convergence at points
1.749 
3.489 
1.750 
3.490 

4.544 
9.593 
4.542 
9.575 

2.086 
9.398 
2.026 
7.835 

Spatial conv. 
Now, most of the errors are because of the spatial discretization, and we can observe that the numerical spatial convergence rates are in the range 1.5–2.7. They suggest that the numerical convergence rate is 2 as it was expected from the previous theoretical analysis.
In
Exact and numerical solutions in the right triangle
In this paper, for the first time, ESERK schemes are proposed to solve nonlinear partial differential equations (PDEs) in right triangles. These codes are explicit, they do not require to solve very large systems of linear nor nonlinear equations at each step. It is demonstrated that such type of codes are able to solve nonlinear PDEs in right triangles. They keep the order of convergence and the absolute stability property under certain conditions. Hence, this paper opens a new line of research, because this new approach will allow, in the future, to solve nonlinear parabolic PDEs with stabilized explicit Runge–Kutta schemes in complex domains, that would be decomposed in rectangles and right triangles.
Additionally, we consider that this procedure can be extended to tetrahedron and other simplixes for the solution of multidimensional nonlinear PDEs in complex regions in ℝ^{n}.
The datasets generated for this study are available on request to the corresponding author.
The author confirms being the sole contributor of this work and has approved it for publication.
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.