Fast meshfree methods for nonlinear radiation diffusion equation

doi:10.1016/j.amc.2022.127571

Applied Mathematics and Computation

Volume 438, 1 February 2023, 127571

https://doi.org/10.1016/j.amc.2022.127571 Get rights and content

Highlights

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In this paper, we used Kansa’s method to solve nonlinear radiation diffusion problems. The full-implicit discretization is adopted in time, and Wendland’s radial basis function is used in space. We presented two new algorithms, DL and SPI for 1D and 2D radiation diffusion problems. Both of them are efficient and accurate. Numerical experiments show that the presented algorithms can obtain small error and good numerical results for nonlinear problems. Moreover, SPI algorithm is better than DL algorithm for 1D and 2D case with the lower errors and small iteration numbers when interior points N is large.

Abstract

Radiation diffusion is a phenomenon of interest in the field of astrophysics, inertial confinement fusion and so on. Since it is modeled by nonlinear equations that are usually solved in complex domains, it is difficult to solve by means of finite element method and finite difference method and so on. In the paper, we will provide a kind of new fast meshfree methods based on radial basis functions. At first, the part of diffusion terms for 1D and 2D radiation diffusion equations are linearized directly on time to form the new implicit schemes, and Kansa’s non-symmetric collocation method with the compactly supported radial basis function is used to solve the radiation diffusion problem. Second, the successive permutation iterative algorithms for full-implicit discretization on time are constructed furtherly, which are more efficient than the former algorithm. In the end, the accuracy and efficiency of the presented algorithms are verified by 1D and 2D numerical experiments. The new meshfree methods not only avoid the complexity of mesh generation, but also solve the radiation diffusion problem with high accuracy.

Introduction

Radiation hydrodynamics has a wide range of applications in inertial confinement fusion (ICF), magnetic confinement fusion, geology astrophysics and so on. It mainly studies the effect of radiation transmission on fluid motion. Radiation transport describes the propagation of radiation in matter and its interaction with matter, which can be described in terms of the radiation diffusion equations. During the transmission in space, radiation is absorbed and emitted, which leads to a strongly nonlinear and strongly coupled process. Therefore, the radiation diffusion equation has the characteristics of “multi-media” and “nonlinear”. So it is very challenging to choose an appropriate numerical simulation method to solve this kind of equations.

In recent years, many scholars have done a lot of researchs on the numerical methods of radiation diffusion equation. Knoll et al. [4], [5] proposed the Jacobian-free Newton–Krylov method to solve the radiation diffusion problem, which is based on the Krylov subspace method. Mousseau et al. [12] proposed preconditioners for nonequilibrium radiation diffusion equations and proved their efficiency. Kang[6] constructed a nonconforming finite element method on unstructured meshes and solved discrete nonlinear equations by using Newton and Picard linearization methods, and the advantages of unstructured $P_{1}$ nonuniform finite element multigrid method are also studied. A simple iterative method is proposed in Olson [14], which can produce very exact second-order solution for coupled radiation and material conduction problems. Sheng et al. [15] established a monotone finite volume scheme on arbitrary polygonal meshes for the non-equilibrium radiation diffusion problem in multi-media, and kept the positivity of the solution. Yuan et al. [18], [19] also constructed a nonlinear iterative method for nonlinear parabolic equation, which proved the effectiveness of the method; and some calculation methods for solving radiation diffusion problems are given in Yuan et al. [20]. Mavriplis[13] uses the traditional finite element method to solve the radiation diffusion equation, and uses the multigrid method to solve the discrete nonlinear equations. In addition, Two kinds of finite volume methods are proposed in Zhao et al. [22], both of which are monotonic. Zhang et al. [23] proposed discontinuous finite element method. The lattice Boltzmann method is used to solve the single-temperature radiation diffusion equation in Wang [16], and the second order accuracy is verified. In order to overcome the shortcomings of the difference scheme, Li et al. [8] constructed a class of meshless methods; as long as the appropriate set of adjacent points of each node is selected, the node information can be used to calculate smoothly. This method solves the two-dimensional three-temperature heat conduction equation on the irregular quadrilateral grid, and obtains ideal numerical results. The element free Galerkin (EFG) method is introduced to solve the heat conduction problem in Kong and Li [7], the time derivative is discretized by using the $θ$ -weighted method. The numerical results show that the EFG method is effective. A new meshless method called generalized finite difference method (GFDM) is used to solve the two-dimensional unsteady heat conduction problem in Xu et al. [17], which is based on Taylor series expansion and weighted least square theory, and transformed the original partial differential equation into an algebraic equations without grid division and numerical integration.

We know that the time-implicit method is commonly used to solve the radiation diffusion equation since it is stable unconditionally. Zhang [24] uses the fully implicit scheme to solve the diffusion equation, which has good stability. Yue et al. [21] construct the Picard-Newton iterative method, which uses the fully implicit scheme to discretize the time, and is applied to solve the non-equilibrium radiation diffusion problems. This method can significantly improve the solving efficiency and achieve the expected results. In the process of radiation transmission in space, the influence of fluid motion should be considered. In the calculation of multi-media Lagrange radiation hydrodynamics, the mesh will deform with fluid motion, so the robustness problem of discrete scheme for the radiation diffusion equation on the large deformation grid is always complicated. Compared with the classical methods such as finite difference method, finite element method, finite volume method and so on, meshless method does not need to generate mesh in solver domain, and radial basis function meshless method is a proper method to study the radiation diffusion problem. The application of radial basis function not only avoids the huge workload of mesh generation, but also avoids the limitation of dimension. Therefore, this paper proposes a new meshless method with fully implicit discretization on time for solving the radiation diffusion equation, which is called Kansa’s non-symmetric collocation method, namely the Kansa’s method [1], [2]. Liu et al. [9], [10], [11] solve nonlinear elliptic equations by this method, effective numerical results are obtained.

We will use the Kansa’s method to solve the one-dimensional(1D) and two-dimensional(2D) radiation diffusion equations in the paper. At first the nonlinear diffusion terms in the radiation diffusion equation can be split into two parts. One linearized method is to take the diffusion coefficient and part of temperature $T$ linearized directly on time; the other is further to construct nonlinear iterative scheme on fully implicit discretization on time. Second, the Kansa’s method is used to solve the schemes constructed by the above two linearization methods. In the end, two numerical examples of 1D and 2D radiation diffusion equations are given to examine the accuracy and superiority of new methods. The new methods not only avoid the complexity of mesh generation, but also have good adaptation since the problem of nonlinearity and deformation mesh in fluid motion are solved well.

The rest content of paper is outlined as follows: Section 2 introduces the concepts of radial basis function, meanwhile gives the radial basis function and its derivatives, while Wendland’s $C^{6}$ function is used as example; Two new linearized schemes and meshfree methods for solving 1D and 2D radiation diffusion problems are provided in Sections 3 and 4; Finally some numerical examples are given to examine the new methods in 5 Numerical experiments, 6 Conclusions reports the conclusion of this paper.

Section snippets

Auxiliary tools from analysis

We introduce a necessary concept related to radial basis function in this section.

Definition 2.1

A multivariate function $Φ : R^{s} \to R$ is called radial provided there exists a univariate function $φ : [0, \infty) \to R$ such that $Φ (x) = φ (r)$ with $r = ∥ x ∥$ . Here $∥ \cdot ∥$ is usually the Euclidean norm. Then the radial basis functions are defined by translation transformation $Φ_{j} (x) = φ (∥ x - ξ_{j} ∥)$ for any fixed center $ξ_{j} \in R^{s}$ .

A now very popular non-symmetric method for the solution of elliptic PDEs with radial basis functions was suggested by Kansa

1D radiation diffusion problem

Consider 1D radiation diffusion equation $c_{v} \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} (K (T) \frac{\partial T}{\partial x}) + S, x \in Ω = (0, L), t > 0,$ where $c_{v}$ is specific heat, $S$ is source item and $K (T)$ is diffusion coefficient. With $T (x, 0) = T_{0} (x)$ is the initial temperature, and the boundary condition is ${T |}_{\partial Ω} = g (x, t), x \in \partial Ω (x = 0 o r x = L) .$

The Eq. (1) can be written as $c_{v} \frac{\partial T}{\partial t} = K^{^{'}} (T) \frac{\partial T}{\partial x} \frac{\partial T}{\partial x} + K (T) \frac{\partial^{2} T}{\partial x^{2}} + S .$

We use the radial basis function $φ_{j} (x) = φ_{3, 3} (∥ x - ξ_{j} ∥) (j = 1, 2, \dots, N)$ to approximate the exact solution $T (x, t_{n})$ , namely $T^{n} (x) = \sum_{j = 1}^{N} c_{j}^{n} φ_{j} (x),$ where ${c_{j}^{n}}_{j = 1}^{N}$ are the unknown coefficients

2D radiation diffusion problem

Consider 2D radiation diffusion equation $c_{v} \frac{\partial T}{\partial t} = \nabla \cdot (K (T) \nabla T) + S, (x, y) \in Ω = {[0, L]}^{2}, t > 0,$ with $T (x, y, 0) = T_{0} (x, y)$ is the initial temperature, and the boundary condition is ${T |}_{\partial Ω} = g (x, y, t), (x, y) \in \partial Ω .$ The Eq. (17) can be split as $c_{v} \frac{\partial T}{\partial t} = K^{^{'}} (T) (\frac{\partial T}{\partial x} \frac{\partial T}{\partial x} + \frac{\partial T}{\partial y} \frac{\partial T}{\partial y}) + K (T) (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}}) + S .$

We use $T^{n} (x, y)$ to approximate the exact solution $T (x, y, t_{n})$ by using the radial basis function $φ_{j} (x, y) = φ_{3, 3} (∥ (x, y) - ξ_{j} ∥) (j = 1, 2, \dots, N)$ , namely $T^{n} (x, y) = \sum_{j = 1}^{N} c_{j}^{n} φ_{j} (x, y),$ where ${c_{j}^{n}}_{j = 1}^{N}$ are the unknown coefficients to be determined. Then, the

Numerical experiments

In this section, numerical experiments for 1D and 2D radiation diffusion problems are given. The Wendland’s $C^{6}$ radial basis functions are used to solve the problems. Three errors are defined as follows:

(1)
The iteration error is commonly noted as $E_{1}^{n + 1, (s + 1)} = {∥ T^{n + 1, (s + 1)} - T^{n + 1, (s)} ∥}_{2} < σ,$ where $σ$ is a small positive number. In the following experiment, we take $σ = 10^{- 8}$ to be as the errors control.
(2)
The discrete $L^{2}$ -error is $L^{2} -error = \sqrt{\frac{1}{M} \sum_{k = 1}^{M} {[T^{n} (ξ_{k}) - T (ξ_{k}, t_{n})]}^{2}},$ where $ξ_{k}$ are the evaluation points.
(3)
The absolute error

Conclusions

In this paper, we use Kansa’s method to solve nonlinear radiation diffusion problems. The full-implicit discretization is adopted in time, and Wendland’s $C^{6}$ function is used in space. We present two new algorithms, DL and SPI for 1D and 2D radiation diffusion problems. Both of them are efficient and accurate. Numerical experiments show that the presented algorithms can obtain small $L^{2}$ -error and good numerical results for nonlinear problems. Moreover, SPI algorithm is better than DL algorithm

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Cited by (0)

^☆: The work is supported by the Natural Science Foundations of China (Nos. 12202219, 12061057), and the Fourth Batch of the Ningxia Youth Talents Supporting Program (No. TJGC2019012).

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Fast meshfree methods for nonlinear radiation diffusion equation☆

Highlights

Abstract

Introduction

Section snippets

Auxiliary tools from analysis

1D radiation diffusion problem

2D radiation diffusion problem

Numerical experiments

Conclusions

Comput. Math. Appl.

J. Quant. Spectrosc. Radiat. Transf.

J. Quant. Spectrosc. Radiat. Transf.

Math. Probl. Eng.

J. Comput. Phys.

Numer. Linear Algebra Appl.

SIAM J. Sci. Comput.

Meshfree Approximation Methods with MATLAB

Kansa RBF method for nonlinear problems

Int. J. Comput. Methods Exp. Meas.

P1 nonconforming finite element multigrid method for radiation transport

SIAM J. Sci. Comput.

Element free Galerkin method to solve heat equation (in Chinese)

J. Comput. Appl.

A class of meshless methods for heat conduction equations

Chin. J. Comput. Phys.