# Radial basis function ENO and WENO finite difference methods based on the optimization of shape parameters

## Abstract

We present adaptive finite difference ENO/WENO methods by adopting infinitely smooth radial basis functions (RBFs). This is a direct extension of the non-polynomial finite volume ENO/WENO method proposed by authors in [4] to the finite difference ENO/WENO method based on the original smoothness indicator scheme developed by Jiang and Shu [9]. The RBF-ENO/WENO finite difference method slightly perturbs the reconstruction coefficients with RBFs as the reconstruction basis and enhances accuracy in the smooth region by locally optimizing the shape parameters. The RBF-ENO/WENO finite difference methods provide more accurate reconstruction than the regular ENO/WENO reconstruction and provide sharper solution profiles near the jump discontinuity. Furthermore the RBF-ENO/WENO methods are easy to implement in the existing regular ENO/WENO code. The numerical results in 1D and 2D presented in this work show that the proposed RBF-ENO/WENO finite difference method better performs than the regular ENO/WENO method.

keywords Essentially non-oscillatory method, Weighted essentially non-oscillatory method, Radial basis function interpolation, Finite difference method.

## 1 Introduction

We are interested in solving hyperbolic problems that could contain discontinuous solutions in time. High order numerical approximations of such discontinuous solutions may suffer from the Gibbs phenomenon and the approximations are highly oscillatory, e.g. spectral approximations of discontinuous functions [8]. High order essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) methods are one of the most powerful methods that solve hyperbolic problems with the Gibbs oscillations much reduced [6, 9, 13]. In [4], we developed non-polynomial ENO and WENO methods based on the non-polynomial reconstruction for the finite volume approximations. The original ENO and WENO methods are based on the polynomial interpolation and its convex combination for the final reconstruction at the cell boundaries [13]. Thus once the number of cells that are participating in the interpolation is fixed, the order of accuracy is also fixed. In [4], we proposed to use non-polynomial bases such as the radial basis functions (RBFs) to improve the fixed order of accuracy of the final reconstruction. The underlying idea is to use non-polynomial functions as the interpolation basis that contain a free parameter. The value of the introduced free parameter is adaptively determined by the local solutions so that the local accuracy is improved and higher order of accuracy than the polynomial order can be achieved. This improvement was made by making the leading error term vanish by the free parameter or at least vanish to a certain order. If the leading error term vanishes, the accuracy is improved particularly when the solution is smooth. In [4], it was shown that the ENO/WENO methods with the non-polynomial basis functions improve the accuracy significantly if the problem is smooth. This paper is a direct extension of our previous work for the finite volume ENO/WENO methods to the finite difference ENO/WENO methods. The formulation of the RBF finite difference ENO/WENO methods is basically same as the finite volume methods, while it is more easier to implement to higher-dimensional problems. In this paper we show that the RBF finite difference ENO/WENO methods work similarly as the RBF finite volume methods and provide supporting numerical examples.

For the non-polynomial basis, we use the multi-quadric (MQ) RBFs. We note that the hybrid of the RBF method and the WENO method is not new. For example, in [1] a polyharmonic spline was used to solve PDEs on the unstructured grid with the WENO method. The RBF WENO method in [1] took advantages of both the RBF and WENO methods. That is, the hybrid method utilizes the meshless feature of the RBFs, which is beneficial for the unstructured grid. It also took the advantage of the non-oscillatory feature of the WENO method, which efficiently handles discontinuous solutions. However, as the polyharmonic spline was used as the RBF basis, the shape parameter was not free but fixed. In this paper, we are more interested in improving the accuracy rather than the efficiency dealing with the unstructured mesh. In order to improve local accuracy, we adopt infinitely smooth RBFs such as the MQ-RBFs, which contain the free parameter, so-called the shape parameter. By exploiting the shape parameter, we improve the local accuracy. It would be an interesting project to develop a method which utilizes both the meshless feature and the shape parameter of the RBFs with the WENO method on the unstructured grid. We only focus on the accuracy in this paper.

We restrict our discussion to RBFs having only one shape parameter although it would be possible to have multiple shape parameters. One of the advantages of using the shape parameter is that the polynomial interpolation is a limit case of the RBF interpolation. That is, the RBF interpolation becomes equivalent to the polynomial interpolation if the shape parameters vanish [10, 11]. In this paper, we follow the similar procedure in [4] for the finite difference ENO/WENO reconstruction and solve several hyperbolic problems. Since the shape parameters are optimized using all the given information within the given stencil, the method would be oscillatory if the stencil contains discontinuities. To prevent such oscillations, we adopt the monotone polynomial interpolation by measuring the local maxima. To switch back to the regular ENO/WENO method, we only need to have the shape parameter vanish, which makes it easy to implement the RBF-ENO/WENO method in the existing ENO/WENO code.

The paper is composed of the following sections. In Section 2, we briefly explain the RBF interpolation and the optimization of the shape parameter. In Section 3, we explain the regular ENO/WENO finite difference method followed by Section 4 where we compare the polynomial interpolation with the RBF interpolation. The interpolation coefficients for and are provided. In Section 5, we provide the numerical examples. For the numerical experiments for discontinuous problems, we first use the monotone polynomial interpolation method for the prevention of oscillations. Then we present the numerical examples for both linear and nonlinear problems and for both scalar and system problems in 1D and 2D. In Section 6, we provide a brief conclusion and our future research.

## 2 RBF Interpolation and optimization of the shape parameter

We briefly explain the RBF interpolation in one space dimension. Suppose that for a domain , a data set is given where the centers are the coordinates and are the function values of the unknown function at . The RBF interpolation is given by a linear combination of the radial basis function, . The kernel at is a function of the radial distance between and the center and the shape parameters , i.e. where is the distance between and . The Euclidean norm is usually used for . Then the RBF interpolation of based on data points, , is given by

(1) |

where are the expansion coefficients to be determined. Here we note that one can add a low order polynomial term in the right hand side of the above equation to make the interpolation consistent up to a certain degree. In this paper, however, we do not include the low order polynomial term in the RBF interpolation. Using the interpolation condition , the expansion coefficients are determined by the linear system

(2) |

where , and is the element of the interpolation matrix . If is one of the piecewise smooth RBFs with no shape parameter involved, we will obtain the RBF interpolation explicitly after we solve the linear system. However, if is one of the infinitely smooth RBFs, we still need to determine or pre-assign the value of the shape parameter. One can have shape parameters fixed globally as a constant. Or one can optimize them so that the accuracy of the interpolation is improved. There are various kinds of RBFs [3]. Among those, we use the multi-quadric (MQ) RBF given by

(3) |

One could use a different form of the MQ-RBF such as , but this form yields a more complicated algebraic expression for the RBF-ENO/WENO formulation than (3).

In [3], it has been discussed that the RBF interpolation may yield spectral convergence. In practice, such a fast convergence depends on several conditions such as the optimization of the shape parameters and the data structure. In this paper, we focus on the optimization of the shape parameter to enhance the overall accuracy. This optimization procedure is the key element of the RBF-ENO/WENO formulation. To explain the optimization procedure below, we first assume that the center set forms a uniform grid and consider the case of . In the following section, similar analysis will be performed for the RBF-ENO/WENO formulation. It is straightforward to extend the same idea to larger values of .

Let , and with . Let denote the function value of at . For example,

where and are to be computed according to (2).

We evaluate the error function at . With the fixed and the obtained and from (2), the Taylor series of in terms of is given by

(4) |

where the superscript denotes the second derivative in . The Taylor series of around yields a similar result but the cubic order term appears as below

(5) |

From (4) and (5), we know that it is possible to obtain the th or rd order accuracy if we choose the shape parameter as

Thus the RBF interpolation can achieve higher convergence than the second order expected by the polynomial interpolation. Or at least we know that there exists that makes the leading error term vanish and helps the interpolation to achieve higher order accuracy. Note, however, that since and are the only given information about the unknown function , the exact value of at and and at are not available. The key idea for the construction of the RBF-ENO/WENO method is that we use the approximation of those values to a certain order based on the function values at the given cells so that we can still obtain the improved order of accuracy. Table 1 shows the adaptation condition of that makes the leading error term at vanish when expanded in the Taylor series around for different values of with centers . The table shows the leading interpolation errors with different when the interpolation is obtained as the polynomials (, Polynomial order) and RBFs when the adaptation is applied (RBF order). As shown in the table the error of the RBF interpolation is of for because of the symmetry and of for the rest while the error of the polynomial interpolation is of for every .

N | Adaptation condition for | Polynomial order | RBF order |
---|---|---|---|

2 | |||

3 | |||

4 | |||

5 |

## 3 1D Finite difference ENO/WENO method

Given a uniform grid with number of points

for the -th cell and uniform grid spacing as , define the cell center

The semi-discretized form of the 1D hyperbolic equation

is a system of ordinary differential equations

(6) |

where is a numerical approximation to and is the flux function. While the finite volume method reconstructs the cell boundary values of the solution, the finite difference method seeks the reconstruction of the flux function at the cell boundaries. Define the numerical flux function for each cell

Differentiate both sides and evaluate them at to obtain

Therefore, (6) becomes

(7) |

where , . Given grid values of

For the -th order ENO reconstruction, we choose the stencil based on cells to the left and cells to the right including such that

Define as the stencil composed of those cells including the cell

(8) |

Define a primitive function such that

(9) |

where the lower limit in the integral can be any cell boundary [12]. By the definition of in (9), it is obvious that . Then for , is given by the linear sum of cell averages

The regular ENO method constructs the polynomial interpolation of based on , while the RBF-ENO method constructs the RBF interpolation of . Suppose is some interpolation for such that

(10) |

Then is the function we seek to approximate where

(11) |

Then the numerical flux function is given by the linear combination of the given flux value at each point

where are fixed constants for the ENO method but are variables for the RBF-ENO method.

The WENO method seeks the reconstruction as a convex combination of all possible ENO reconstructions. Suppose the stencil (8) produces different reconstructions to the value . Then the WENO reconstruction would take the convex combination of all :

(12) |

where

with

Here are the polynomial expansion coefficients and is introduced to avoid the case that the denominator becomes zero usually taken as . are the “smoothness indicators” of the stencil . In this work, we use the smoothness indicators developed in [9]. The usage of different smoothness indicators found in other WENO variations such as the WENO-Z method [2], WENO-M method [7] or WENO-P method [5] will be considered in our future work.

## 4 1D reconstruction based on RBFs

The 1D and 2D finite difference RBF-ENO method is a direct extension of the finite volume RBF-ENO method [4]. We only replace the cell averages with the nodal flux function . First we consider the case of , that is, two cells are used for the reconstruction. For this case, three flux values, , and are available. To reconstruct the boundary values of and , we use either or . Which flux values should be used is decided by the Newton’s divided difference method [12]. Assume that we decided to use from the Newton’s divided difference. We only show the reconstruction at cell boundary . The reconstruction at can be achieved in the same manner. Based on the definition of in (9) we have

### 4.1 Polynomial interpolation for regular ENO

The polynomial interpolation is given as

Let

Then the expansion coefficients are given by solving the linear system . After taking the first derivative of and plugging it in , we obtain the numerical flux function

(13) |

Expanding around in the Taylor series yields

(14) |

The first term in the right hand side of (14) is the exact value of at . Thus we confirm that (13) is a nd order reconstruction of .

### 4.2 Multi-Quadratic RBF interpolation

Now we consider the RBF reconstruction of . The MQ-RBF interpolation for is given as

Accordingly the interpolation matrix is given by

We take the first derivative of to obtain the numerical flux at

(15) |

Expanding around in the Taylor series yields

(16) | |||||

Thus we observe that (15) is at least nd order accurate to . If we take the value of as below

(17) |

then the nd order error term vanishes and we obtain a th order accurate approximation. We notice that the coefficients of and in (15) have complicated forms involving a calculation of square roots. This is common among infinitely smooth RBF interpolations because of the existence of undetermined shape parameters. If we fix the shape parameter as a constant, the reconstruction would have simpler forms as the piecewise smooth RBF interpolation, but lose the ability to improve accuracy. In order to simplify the form, we expand the right hand side of (15) as below

(18) | |||||

Ignoring all the high order terms in (18) yields

(19) |

Expanding and in terms of we get

(20) | |||||

Thus (19) yields the same th order accuracy equivalently to (15) when (17) is used.

Now we consider the evaluation of in (17) to achieve higher order accuracy than the nd order. To explain this, we use the reconstruction at as an example. The case of can be done in the same manner. The idea is that although the ENO method chooses either or for the final reconstruction, all three flux function values are available when the Newton’s divided difference method is applied. Thus we calculate and based on and . Construct the Lagrange interpolation of based on

The second derivative of is approximated by the third derivative of , which is given by

Then by plugging the above approximations of and into (17), we approximate the optimal value of as below

(21) |

Note that can be a complex number because can be a negative real number. Since is used instead of , the final reconstruction is still done with the real operation. If we replace in (16) with that in (21), we get the following

Thus if is chosen as in (21), the reconstruction (19) still yields a rd order accuracy although we do not completely make the nd order term in (20) vanish.

### 4.3 Reconstruction coefficients

Tables 3 and 3 show the reconstruction coefficients for and , respectively for the MQ RBF-ENO reconstruction. These are the same as for the finite volume method provided in [4]. The difference is that the flux function is used for the calculation of the optimal value of , while the cell average of the solution is used for the finite volume method. As we see in these tables, the RBF-ENO reconstruction is a perturbed reconstruction of the polynomial-based ENO interpolation. We also observe that the coefficients for are not consistent while those for are consistent. Here note that when the shape parameter vanishes, i.e. when , the MQ-RBF reconstruction is reduced into the polynomial reconstruction. For this equivalence known as the polynomial limit [10, 11], one can easily switch the RBF-ENO to the regular ENO and it is easy to convert the existing ENO code into the RBF-ENO code.

Table 4 shows the leading error terms we want to make vanish as in (16) to achieve (k+1)th order of convergence when cells are used. The adaptation condition becomes complicated for the approximation of when the value of becomes large. We leave efficient evaluation methods of for larger values of for our future work.

k | r | j=0 | j=1 |

-1 | |||

2 | 0 | ||

1 | |||

k | r | j=0 | j=1 |

-1 | |||

2 | 0 | ||

1 | |||

k | r | j=0 | j=1 | j=2 |

-1 | ||||

3 | 0 | |||

1 | ||||

2 | ||||

k | r | j=0 | j=1 | j=2 |

-1 | ||||

3 | 0 | |||

1 | ||||

2 | ||||

k | coefficient in the leading error term |
---|---|

2 | |

3 | |

4 | |

5 |

## 5 Numerical experiments

For the numerical example, we compare the developed method with the original ENO/WENO method denoted by the ENO/WENO-JS developed by Jiang and Shu [9]. For the WENO method we use the smoothness indicators developed in [9] and denote our proposed RBF-WENO method as the RBF-WENO-JS method. One can adopt other developments rather than the original WENO method for the RBF-WENO method such as the WENO-Z method [2], which will be investigated in our future research. For the numerical experiments, we use the Lax-Friedrich flux scheme with the rd order TVD Runge-Kutta method [14] for the time integration.

### 5.1 Polynomial limit and non-oscillatory reconstruction

The ENO method uses adaptive stencil to avoid the reconstruction across the possible discontinuity. The RBF-ENO method, however, as the WENO reconstruction, utilizes all available flux functions to optimize the shape parameter within the stencil . Thus the final reconstruction can be oscillatory. To prevent the oscillation and recover the ENO property, one can easily switch the RBF-ENO reconstruction to the ENO reconstruction by using the polynomial limit of RBFs. That is, if the shape parameter vanishes, the RBF reconstruction becomes equivalent to the polynomial reconstruction and becomes the regular ENO reconstruction. In [4] the monotone polynomial method was proposed to use the polynomial limit whenever it is necessary. For the finite difference RBF-ENO method, we follow the same procedure. The difference is that for the local extrema we use the flux function given at each instead of the solution .

The optimized value of in (17) involves the second derivative of , e.g. . For , there are three cells involved and only one is to be computed while for , three possible values of are to be computed. For the illustration consider the case of . For , the same procedure is straightforwardly applied for each ENO block. For , the whole interval of in the stencil is and . In this interval, is approximated by the second order polynomial and its local extrema exists at . Without loss of generality, let

(22) |

If exists inside the given stencil, is not a monotone function. We use the polynomial limit if exists inside the interval of . Thus the polynomial limit condition is given by

(23) |

### 5.2 1D Numerical examples

#### 5.2.1 Advection equation I: smooth initial condition

First we consider the linear scalar equation in

(24) |

with the initial condition

(25) |

and the periodic boundary condition. The CFL condition is given by with . Tables 6 and 6 show the and errors for each method at the final time with and , respectively. As shown in the tables, for , the RBF-ENO method has almost rd order convergence while the regular ENO method yields nd order of accuracy or less. The RBF-ENO is also better performed than the WENO-JS in terms of accuracy while the rates of convergence are similar. The RBF-WENO-JS with is also better than WENO-JS both in accuracy and convergence. We have similar results for . For the RBF-ENO method yields convergence higher than rd order and the RBF-WENO-JS method higher than th order while the regular ENO method is about rd order and the WENO-JS is about th order accurate.

Method | N | error | order | error | order | error | order |
---|---|---|---|---|---|---|---|

10 | 1.11E-1 | – | 1.40E-1 | – | 2.22E-1 | – | |

20 | 4.61E-2 | 1.2689 | 5.32E-2 | 1.3994 | 9.43E-2 | 1.2375 | |

ENO | 40 | 1.37E-2 | 1.7458 | 1.79E-2 | 1.5749 | 4.03E-2 | 1.2236 |

k = 2 | 80 | 3.81E-3 | 1.8530 | 5.69E-3 | 1.6492 | 1.68E-2 | 1.2614 |

160 | 1.02E-3 | 1.8901 | 1.80E-3 | 1.6599 | 6.91E-3 | 1.2849 | |

320 | 2.71E-4 | 1.9224 | 5.69E-4 | 1.6613 | 2.81E-3 | 1.2964 | |

10 | 1.79E-2 | – | 2.35E-2 | – | 4.24E-2 | – | |

20 | 2.48E-3 | 2.8540 | 2.65E-3 | 3.1473 | 3.62E-3 | 3.5495 | |

RBF-ENO | 40 | 3.17E-4 | 2.9673 | 3.43E-4 | 2.9488 | 4.79E-4 | 2.9207 |

k = 2 | 80 | 4.05E-5 | 2.9704 | 4.42E-5 | 2.9592 | 6.22E-5 | 2.9434 |

160 | 5.17E-6 | 2.9702 | 5.60E-6 | 2.9810 | 7.97E-6 | 2.9652 | |

320 | 6.51E-7 | 2.9897 | 7.05E-7 | 2.9892 | 1.00E-6 | 2.9894 | |

10 | 9.13E-2 | – | 1.10E-1 | – | 1.74E-1 | – | |

20 | 2.92E-2 | 1.6453 | 3.26E-2 | 1.7588 | 5.51E-2 | 1.6547 | |

WENO-JS | 40 | 4.81E-3 | 2.5998 | 6.33E-3 | 2.3645 | 1.38E-2 | 2.0037 |

k = 2 | 80 | 6.42E-4 | 2.9066 | 9.43E-4 | 2.7470 | 2.61E-3 | 2.3979 |

160 | 7.79E-5 | 3.0440 | 1.23E-4 | 2.9427 | 3.96E-4 | 2.7183 | |

320 | 9.54E-6 | 3.0291 | 1.53E-5 | 3.0035 | 5.25E-5 | 2.9181 | |

10 | 1.88E-2 | – | 2.38E-2 | – | 4.06E-2 | – | |

20 | 2.60E-3 | 2.8561 | 2.73E-3 | 3.1235 | 4.13E-3 | 3.2995 | |

RBF-WENO-JS | 40 | 3.25E-4 | 2.9969 | 3.57E-4 | 2.9367 | 5.36E-4 | 2.9442 |

k = 2 | 80 | 4.05E-5 | 3.0064 | 4.50E-5 | 2.9871 | 6.75E-5 | 2.9896 |

160 | 5.09E-6 | 2.9910 | 5.63E-6 | 2.9983 | 8.32E-6 | 3.0199 | |

320 | 6.40E-7 | 2.9927 | 7.04E-7 | 3.0000 | 1.00E-6 | 3.0539 |

Method | N | error | order | error | order | error | order |
---|---|---|---|---|---|---|---|

10 | 2.32E-2 | – | 2.54E-2 | – | 3.65E-2 | – | |

20 | 2.79E-3 | 3.0570 | 3.02E-3 | 3.0727 | 4.47E-3 | 3.0314 | |

ENO | 40 | 3.36E-4 | 3.0519 | 3.69E-4 | 3.0360 | 5.47E-4 | 3.0293 |

k = 3 | 80 | 4.12E-5 | 3.0284 | 4.55E-5 | 3.0189 | 6.76E-5 | 3.0167 |

160 | 5.10E-6 | 3.0146 | 5.65E-6 | 3.0099 | 8.53E-6 | 2.9878 | |

320 | 6.34E-7 | 3.0074 | 7.03E-7 | 3.0051 | 1.06E-6 | 3.0082 | |

10 | 1.89E-2 | – | 2.22E-2 | – | 3.45E-2 | – | |

20 | 2.15E-3 | 3.1332 | 2.59E-3 | 3.1022 | 4.47E-3 | 2.9462 | |

RBF-ENO | 40 | 1.51E-4 | 3.8321 | 2.18E-4 | 3.5731 | 5.14E-4 | 3.1218 |

k = 3 | 80 | 8.81E-6 | 4.1023 | 1.57E-5 | 3.7941 | 5.10E-5 | 3.3319 |

160 | 4.83E-7 | 4.1891 | 1.06E-6 | 3.8854 | 4.60E-6 | 3.4706 | |

320 | 2.78E-8 | 4.1205 | 7.30E-8 | 3.8618 | 4.25E-7 | 3.4372 | |

10 | 9.74E-3 | – | 1.13E-2 | – | 1.63E-2 | – | |

20 | 4.01E-4 | 4.6032 | 4.63E-4 | 4.6147 | 7.82E-4 | 4.3822 | |

WENO-JS | 40 | 1.17E-5 | 5.0966 | 1.37E-5 | 5.0804 | 2.45E-5 | 4.9965 |

k = 3 | 80 | 3.55E-7 | 5.0452 | 4.10E-7 | 5.0617 | 7.55E-7 | 5.0196 |

160 | 1.12E-8 | 4.9825 | 1.27E-8 | 5.0100 | 2.33E-8 | 5.0180 | |

320 | 3.80E-10 | 4.8849 | 4.25E-10 | 4.9022 | 6.94E-10 | 5.0692 | |

10 | 6.53E-3 | – | 7.90E-3 | – | 1.16E-2 | – | |

20 | 1.22E-5 | 5.7476 | 1.50E-4 | 5.7155 | 2.72E-4 | 5.4183 | |

RBF-WENO-JS | 40 | 2.58E-6 | 5.5559 | 3.24E-6 | 5.5380 | 8.04E-6 | 5.0796 |

k = 3 | 80 | 7.51E-8 | 5.1050 | 8.72E-8 | 5.2129 | 2.17E-7 | 5.2082 |

160 | 2.35E-9 | 4.9964 | 2.63E-9 | 5.0497 | 5.58E-9 | 5.2833 | |

320 | 7.39E-11 | 4.9920 | 8.22E-11 | 5.0014 | 1.46E-10 | 5.2581 |

#### 5.2.2 Advection equation II: non-smooth initial condition

We consider the same advection equation (24) but with the discontinuous initial condition

(26) |

and the boundary condition . Here sgn(x) is the sign function of . Figure 2 shows solutions at by each method with . The top two figures show the solutions with and the bottom two figures with . As shown in the figures, the Gibbs oscillations are not significant and the RBF-ENO and RBF-WENO-JS methods yield almost non-oscillatory solutions. For , the RBF-ENO and RBF-WENO-JS solutions are similar and have sharper solution profiles than the regular ENO and WENO methods. For . The RBF-WENO-JS solution is slightly sharper than the WENO-JS solution. The RBF-ENO solution is sharper than the ENO solution but smoother than the RBF-WENO-JS or WENO-JS solutions.

#### 5.2.3 Burger’s equation

We consider the Burgers’ equation for

(27) |

with the initial condition

(28) |

Tables 8 and 8 show various errors for each case for and , respectively. For , the RBF-ENO and RBF-WENO-JS methods yield much better accuracy than the ENO or WENO-JS methods, e.g. errors. For , the RBF-WENO-JS method yields convergence higher than th order while the WENO-JS is about th order accurate. The RBF-ENO solution yields convergence higher than rd order while the regular ENO solution is about rd order or less accurate.

Method | N | error | order | error | order | error | order |
---|---|---|---|---|---|---|---|

10 | 5.38E-2 | – | 8.24E-2 | – | 1.71E-1 | – | |

20 | 1.98E-2 | 1.4464 | 2.81E-2 | 1.5521 | 5.74E-2 | 1.5716 | |

ENO | 40 | 5.40E-3 | 1.8715 | 8.28E-3 | 1.7628 | 2.12E-2 | 1.4355 |

k = 2 | 80 | 1.31E-3 | 2.0396 | 2.52E-3 | 1.7183 | 8.85E-3 | 1.2619 |

160 | 3.85E-4 | 1.7690 | 8.13E-3 | 1.6309 | 3.64E-3 | 1.2816 | |

320 | 1.07E-4 | 1.8475 | 2.59E-4 | 1.6519 | 1.49E-3 | 1.2928 | |

10 | 4.13E-2 | – | 4.52E-2 | – | 1.16E-1 | – | |

20 | 7.41E-3 | 2.4795 | 1.53E-2 | 1.5817 | 4.63E-2 | 1.3254 | |

RBF-ENO | 40 | 1.08E-3 | 2.7836 | 3.01E-3 | 2.3418 | 1.18E-2 | 1.9714 |

k = 2 | 80 | 1.38E-4 | 2.9656 | 4.22E-4 | 2.8362 | 1.81E-3 | 2.7073 |

160 | 1.55E-5 | 3.1548 | 4.89E-5 | 3.1107 | 2.40E-4 | 2.9135 | |

320 | 1.75E-6 | 3.1415 | 5.54E-6 | 3.1396 | 2.67E-5 | 3.1669 | |

10 | 4.61E-2 | – | 7.23E-2 | – | 1.55E-1 | – | |

20 | 1.27E-2 | 1.8592 | 1.96E-2 | 1.8792 | 4.43E-2 | 1.8056 | |

WENO-JS | 40 | 1.97E-3 | 2.6901 | 3.13E-3 | 2.6493 | 8.18E-3 | 2.4392 |

k = 2 | 80 | 2.99E-4 | 2.7199 | 5.22E-4 | 2.5851 | 1.40E-3 | 2.5473 |

160 | 4.32E-5 | 2.7893 | 7.89E-5 | 2.7252 | 2.49E-4 | 2.4893 | |

320 | 5.75E-6 | 2.9100 | 1.07E-5 | 2.8769 | 3.68E-5 | 2.7572 | |

10 | 2.50E-2 | – | 3.69E-2 | – | 7.02E-2 | – | |

20 | 6.64E-3 | 1.9158 | 1.57E-2 | 1.2310 | 4.88E-2 | 0.5227 | |

RBF-WENO-JS | 40 | 1.05E-3 | 2.6662 | 3.06E-3 | 2.3610 | 1.28E-2 | 1.9376 |

k = 2 | 80 | 1.30E-4 | 3.0062 | 4.03E-4 | 2.9230 | 1.70E-3 | 2.9098 |

160 | 1.50E-5 | 3.1181 | 4.62E-5 | 3.1254 | 2.15E-4 | 2.9772 | |

320 | 1.73E-6 | 3.1128 | 5.29E-6 | 3.1286 | 2.45E-5 | 3.1374 |

Method | N | error | order | error | order | error | order |
---|---|---|---|---|---|---|---|

10 | 2.00E-2 | – | 3.63E-2 | – | 7.95E-2 | – | |

20 | 4.89E-3 | 2.0328 | 7.77E-3 | 2.2243 | 1.57E-2 | 2.3449 | |

ENO | 40 | 8.68E-4 | 2.4953 | 1.50E-3 | 2.3751 | 4.93E-3 | 1.6676 |

k = 3 | 80 | 1.25E-4 | 2.7961 | 2.21E-4 | 2.7622 | 6.38E-4 | 2.9499 |

160 | 1.84E-5 | 2.7675 | 3.44E-5 | 2.6816 | 1.32E-4 | 2.2697 | |

320 | 2.84E-6 | 2.6918 | 5.68E-6 | 2.5991 | 2.84E-5 | 2.2184 | |

10 | 1.87E-2 | – | 3.48E-2 | – | 7.70E-2 | – | |

20 | 4.05E-3 | 2.2024 | 6.22E-3 | 2.4851 | 1.44E-2 | 2.4187 | |

RBF-ENO | 40 | 6.21E-4 | 2.7069 | 1.17E-3 | 2.4052 | 3.55E-3 | 2.0213 |

k = 3 | 80 | 1.22E-4 | 2.3453 | 3.00E-4 | 1.9658 | 1.53E-3 | 1.2130 |

160 | 1.03E-5 | 3.5705 | 3.00E-5 | 3.3226 | 1.91E-4 | 3.0040 | |

320 | 8.23E-7 | 3.6436 | 2.76E-6 | 3.4444 | 2.09E-5 | 3.1898 | |

10 | 1.28E-2 | – | 2.21E-2 | – | 4.75E-2 | – | |

20 | 2.67E-3 | 2.2595 | 4.97E-3 | 2.1570 | 1.49E-2 | 1.6760 | |

WENO-JS | 40 | 2.83E-4 | 3.2361 | 8.51E-4 | 2.5439 | 3.75E-3 | 1.9854 |

k = 3 | 80 |