National Taiwan University

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1 National Taiwan University Meshless Methods for Scientific Computing (Advisor: C.S. Chen, D.L. oung) Final Project Department: Mechanical Engineering Student: Kai-Nung Cheng SID: D9956 Date: Jan. 8

2 The Time-Marching MFS and Time-Marching LMAPS for Solving Klein-Gordon Equations

3 . Introduction In this report, we consider a partial differential equation (PDE) problem called the Klein-Gordon equation, and which is a relativistic version of the Schrodinger equation. It was first found by Schrodinger. The Klein-Gordon equation is the motion equation of a quantum scalar or pseudo scalar field, a field whose quanta are spinless particle. It is second order in time and does not admit a positive definite conserved probability density. With the appropriate interpretation, it does describe the quantum amplitude for finding a point particle in various places, the relativistic wave function, but the particle propagates both forwards and backwards in time []. For solving the Klein-Gordon equation, we first introduce a meshless numerical method called the time-marching method of fundamental solutions (TM-MFS) to obtain the approximate solution of Klein-Gordon equation [, 3]. It combines the Houbolt finite difference (FD) scheme with the method of the particular solutions (MPS) and the method of fundamental solutions (MFS). Implementing the Houbolt FD with the MFS+MPS is easy to eliminate the difficulties of dealing with time variable and the initial conditions in the PDE to form the linear system of equations. In the other hand, we also consider a large time domain problem described by the Klein-Gordon equation. In order to efficiently solve the large problem, we introduce a localized meshless method combined with the Houbolt FD to obtain the approximate solution of Klein-Gordon equation [4], and we call this method the time-marching localized method of approximate particular solutions (TM-LMAPS). In this report, we consider two kinds of -D Klein-Gordon equations which are the linear homogenous and the nonhomogeneous problems, and both of them are solved by using the TM-MFS and TM-LMAPS, respectively.. Definitions of Klein-Gordon equation The -D linear homogenous and the -D nonhomogeneous Klein-Gordon equations are described as below [5]: () The general form of -D linear homogenous Klein-Gordon equation can be written as: u t = a u bu (.) and one of its exact solutions is expressed as follows: u(x, y, t) = e ±μt (Acosλx + Bsinλy), b = a λ μ (.) where a, b, A and B are arbitrary constants. () The general form of -D linear nonhomogeneous Klein-Gordon equation with a source term is expressed as follows:

4 u t = a u bu + v(x, y, t) (.3) and its exact solution can be expressed as follows: u(x, y, t) = l b sin(t b) + sin (t a λ n + b) l cos(λ nx)cos(λ n y) n= a λ n + b (.4) where b > and λ n = πn l. Based on Eq. (.) and (.3), we assume two Klein-Gordon problems, the homogenous and the nonhomogeneous cases, corresponding to the definitions of Klein-Gordon equation above, and then solve them to obtain the approximate solutions by using the global method (TM-MFS) and local method (TM-LMAPS), respectively. Therefore, four examples have been constructed with giving the different requirements which are described as below: Example : the linear homogenous problem solved by the TM-MFS From Eq. (.), we choose a = and b =, and then the linear homogenous Klein-Gordon equation can be written as follows: u t = u + u, (x, y) Ω (.5) The initial conditions are: u(x, y, t) t= = cosx + 3siny (.6) u t (x, y, t) t= = (cosx + 3siny) (.7) and the boundary condition is given by u = e t (cosx + 3siny), (x, y) Ω (.8) The boundary profile is the star shape domain with the Dirichlet boundary condition only and shown in Figure -. Furthermore, the exact solution of Eq. (.5) can be obtained from Eq. (.), and it is expressed as follows: u = e t (cosx + 3siny) (.9) 3

5 Figure - Star shape domain with the Dirichlet boundary condition Example : the linear nonhomogeneous problem solved by the TM-MFS From Eq. (.3), we choose a =, b =, l = and n =, and then the linear nonhomogeneous Klein-Gordon equation can be written as follows: u t = u u + π sint cos πx cos πy, (x, y) Ω (.) The initial conditions are: u(x, y, t) t= = (.) u t (x, y, t) t= = + cos πx cos πy (.) and the boundary conditions are given by u = sint ( + cos πx cos πy ) (x, y) ΩD (.3) u πx πy = *sint ( + cos cos n )+ n (x, y) Ω N (.4) The boundary profile is the Cassini domain with the mixed boundary condition and shown in Figure -. In the figure, Ω D is Dirichlet boundary above the x-axis and Ω N is Neumann boundary below the x-axis. Furthermore, the exact solution of Eq. (.) can be obtained from Eq. (.4), and it is expressed as follows: u = sint ( + cos πx cos πy ) (.5) 4

6 .5.5 Ω D Ω N Figure - Cassini domain with the mixed boundary condition Example 3: the linear homogenous problem solved by the TM-LMAPS We consider a large problem which can be described by Eq. (.5), and its two initial conditions are the same as Eq. (.6) and (.7). We give this problem the mixed boundary condition expressed as follows: u = e t (cosx + 3siny) (x, y) Ω D (.6) u n = [e t (cosx + 3siny)] n (x, y) Ω N (.7) The boundary profile is the Amoeba shape domain and shown in Figure -3. In the figure, Ω D is Dirichlet boundary above the x-axis and Ω N is Neumann boundary below the x-axis. Example 4: the linear nonhomogeneous problem solved by the TM-LMAPS Similarly, we use Eq. (.) to describe the large Klein-Gordon problem, and its two initial conditions and the mixed boundary condition are the same as the description of Example. Its boundary profile we choose is also the Amoeba shape domain shown in Figure -3. Ω D - Ω N - 3 Figure -3 Amoeba shape domain with the mixed boundary condition 5

7 3. Numerical Algorithm In order to simplify time domain problem, both of meshless methods involving the MFS+MPS and LMAPS, which combine with the Houbolt FD method, respectively, are introduced in this section. The Houbolt FD method is used to discretize the time domain and to eliminate the time variable in the PDE. The TM-MFS is used for solving general problems and TM-LMAPS is used for solving large problems. The numerical algorithms of them are described in the following sub-sections. 3. Time-marching method of fundamental solutions (TM-MFS) For example, a time domain Poisson equation and its boundary condition can be expressed as follows: u c t = u x Ω (3.) B(u) = b(x, t) x Ω (3.) where B is the boundary operator. In order to simplify the time domain problem, the Houbolt FD method is used to discretize the time domain in the PDE. It is an implicit and unconditionally stable time integration method to obtain the first and second order differential terms in time within a small time interval from (n ) t to (n + ) t, and both of terms can be expressed as follows: u n+ t 6 t (un+ 8u n + 9u n u n ) (3.3) u n+ t t (un+ 5u n + 4u n u n ) (3.4) where t is the time interval and u n is u in the n-th time level. Therefore, Eq. (3.) can be re-written by Eq. (3.4) as follow: u n+ = c t (un+ 5u n + 4u n u n ) (3.5) Based on the MFS+MFS, the solution of Eq. (3.5) can be expressed as follows: u n+ = u H n+ + u P n+ (3.6) where u H and u P are the homogenous solution and particular solution, respectively. The homogenous solution can be obtained by the linear combination of fundamental solutions as follows: 6

8 n b u n+ H = α n+ j G x ξ j j= (3.7) where n b is the number of the boundary points, x are the boundary points, ξ are the source points and α is the coefficient of the fundamental solution. In -D problem, G x ξ j can be obtained by: G x ξ j = π ln x ξ j (3.8) On the other hand, the particular solution can be approximated by the radial basis function as follows: n i u P n+ = β j n+ F(r j ) j= (3.9) where n i is the number of the collocation points, F is the radial basis function and β is the coefficient of the basis function. In this report, we choose the compactly supported RBF (CS-RBF) as the basis function (f(r)), and F(r) is the particular solution of f(r) [6]. Both f(r) and F(r) can be expressed as follows: ΔF(r) = f(r) (3.) f(r) = { ( r λ ), r λ, r λ (3.) r 4 6λ F(r) = r3 9λ + r, r λ 4 3λ (3.) { 44 + λ ln (r λ ), r λ Then we can re-write Eq. (3.) and (3.) by the equations mentioned above as the following forms: n i β j n+ [f(r j ) CF(r j )] j= n b α n+ j CG x ξ j j= = C( 5un + 4u n u n ) (3.3) n i β j n+ B[F(r j )] j= n b α n+ j B[G x ξ ] j j= = b(x, t n+ ) (3.4) where C = c Δt and the system of equations is easy to be written as follows: 7

9 [ A A ] [ βn+ A 3 A 4 α n+] = *S b + (3.5) where all sub-matrices of Eq. (3.5) are: A = f(r) CF(r), A = CG x ξ, j A 3 = B[F(r)], A 4 = B[G x ξ ] j S = C( 5un + 4u n u n ) (3.6) In Eq. (3.6), u n and u n can be obtained by using the Euler explicit scheme and they are expressed as follows: u n = I I (x ) Δt I II (x ) u n = I I (x ) Δt I II (x ), n (3.7) After obtaining the coefficients, α n+ and β n+, by solving Eq. (3.5) and (3.6), the approximate solution of Eq. (3.) and (3.) can be calculated by: n i u n+ = β j n+ F(r j ) j= n b + α n+ j G x ξ j j= (3.8) 3. Time-marching localized method of approximate particular solutions (TM-LMAPS) For large problems, they can be solved by implementing the localized method of approximate particular solutions (LMAPS). Using the LMAPS to solve the PDE, we can use any collocation point as the center point to construct the local domain which only involves a few n s nearest neighbor points (i.e. Ω i = {x i k }, k ns) to efficiently calculate the approximate solution u (x i ) with low computational interpolation. By the collocation method, we can assume that the approximate solutions u (x n i ) of n collocation points in the local domains can be expressed as the following linear system: u (x i ) Φ x i x i u (x i ) Φ x i x i : : [ u (x i n )] [ Φ x i n x i Φ x i x i Φ x i x i : Φ x n i x i.... :.. Φ x i x n i Φ x i x n i : Φ x n i x n i ] = [ α α : ] (3.9) α n where α is the unknown coefficient, and we can re-write Eq. (3.9) as follows: 8

10 α = Φ u n (3.) From Eq. (3.9) and (3.), we can re-formulate u (x i ) in each local domain as follows: u (x i ) = Φ (x i )α = Φ (x i )Φ u n = Ψ n (x i )u n (3.) where Ψ n = Φ (x i )Φ = [φ, φ,, φ n ] (3.) From Eq. (3.), we can use the global u N instead of local u n by padding vector Ψ n (x) with zero entries based on the mapping between u n and u N. It follows that u (x i ) = Ψ N (x i )u N (3.3) where Ψ N (x i ) is a sparse vector with N components which is obtained by inserting N n zero into Ψ n (x i ) at the proper places. For solving the time domain problem, i.e. Eq. (3.) and (3.), by using the local domains mentioned above, the domain equation, Eq. (3.), can be re-defined with the Houbolt FD method, i.e. Eq. (3.5), as follows: n f(x i ) = α k ΔΦ n(x i ) k= = c t (un+ 5u n + 4u n u n ) (3.4) and the boundary equations, i.e. Eq. (3.), can be re-defined as follows: n g(x i ) = α k BΦ n(x i ) = b(x i, t n+ ) (3.5) k= where α is defined by Eq. (3.). Therefore, we can extend u n of Eq. (3.4) and (3.5) to u N by: f(x i ) = Ξ N (x i )u N i N i (3.6) g(x i ) = Υ N (x i )u N N i i N (3.7) Eventually, we can use Eq. (3.6) and (3.7) to easily construct the sparse system of equations as follows: 9

11 Ξ N (x ) : Ξ N (x Ni ) Υ N (x Ni+ ) [ : Υ N (x N ) ] [ u (x ) : u (x Ni ) u (x Ni+ ) : u (x N ) ] = f(x ) : f(x Ni ) g(x Ni+ ) : [ g(x N ) ] (3.8) By solving Eq. (3.8), we can obtain the approximate solution u at all given points. In the TM-LMAPS, the MQ is chosen as the basis function so that we have φ(r) = r + (cr ) (3.9) with normalized shape parameter c, and r is the maximum distance from the center point to all neighbor points in the local domain. The particular solutions Φ(r) of φ(r) is Φ(r) = 9 (4(cr ) + r ) r + (cr ) (cr ) ln (cr 3 + r + (cr ) ) (3.3) Furthermore, Eq. (3.9) and (3.3) are used to substitute ΔΦ n(x i ) and Φ n(x i ) in Eq. (3.4) and (3.5). 3.3 Error estimation In this report, three kinds of error estimation are used to measure the accuracy of the approximate solution which is obtained from either the TM-MFS or TM-LMAPS. They are the, AME and ARE, and are described as follows: The (Root Mean Square Error) is defined by: N = N u k u k k= (3.3) the AME (Absolute Maximum Error) is defined by: AME = max( u k u k ) k N (3.3) and the ARE (Absolute Relative Error) is defined by: ARE = max ( u k u k u k ) k N (3.33)

12 where u k is the approximate solution, u k is the exact solution and N is the number of the test points. To measure the accuracy of approximate solution in each example described in Section, the AME is used for the linear homogenous Klein-Gordon problem and the ARE is used for the linear nonhomogeneous Klein-Gordon problem. Furthermore, the is used for both of problems. 4. Numerical Results In Section, four examples have been constructed to describe the linear homogenous and the linear nonhomogeneous Klein-Gordon problems with the different domain and boundary conditions, respectively. In this section, we use the global and the local meshless methods, the TM-MFS and TM-LMAPS described in Sections 3. and 3., to solve the Klein-Gordon equations and to efficiently obtain the accurate approximate solution from each example. Example : the linear homogenous Klein-Gordon problem solved by the TM-MFS In order to solve Eq. (.5) to (.8) with the star shape domain shown in Figure - by the TM-MFS, we can generate the interpolation points distributing inside and outside the star shape domain like the pattern of Figure 4- (left) and the test points shown in Figure 4- (right) for the error estimation of the approximate solution. In Example, several cases with the different input data have been considered and listed in Table 4-. To implement the MFS+MPS, we choose, 3 and 4 quasi-random interior points inside the domain, respectively, boundary points on the boundary and source points around the circle with R = 3.5, 4. and 4.5, respectively. The total,48 uniformly distributed test points are used to test the error of the approximate solution. Furthermore, the time interval including E-, 5E-3 and E-3 are used to discretize the time domain in the TM-MFS, respectively. As shown in Figure 4-, the results of the versus time in each case have been obtained by the TM-MFS. All results show that the keeps linearly descending when the time increases and the accurate approximate solution in each case can be obtained by using a small time interval (i.e. t = E-3). According to the wave motion during the time history shown in Figure 4-3, the initial transient wave will tend to be motionless after t = 5. The exact solution of the linear homogenous Klein-Gordon equation, i.e. Eq. (.9), completely represents the trend of the wave motion shown in Figure 4-3. Due to the wave motion to be motionless in the end, it can be found that the will approach to zero if the time is infinite. Furthermore, we also estimate the results of the AME and in the specific time using 4 interior points, boundary points, source points with R = 4. and t = E-3, and they are shown in Figure 4-4. For example, we obtain that the AME is 4.6 E- and is.69 E- at t =., and AME is 3.67 E-4 and is 8.84 E-5 at t = 5..

13 Table 4- Points and time interval for the TM-MFS in Example Case MPS MFS Interior points Boundary points Source points Source location a R = 4.5 b 3 R = 3.5 c 4 R = 4. Houbolt FD t = E-, 5E-3 and E-3 Test points,4 uniformly distributed points and 6 boundary points Figure 4- Point distribution (blue: interior points, green: boundary points and red: source points) for the TM-MFS in Example (left) and test points (right)

14 - t = E- t = 5E-3 t = E-3 - t = E- t = 5E-3 t = E t (a) interior points, boundary points and source points with R = t (b) 3 interior points, boundary points and source points with R = 3.5 t = E- t = 5E-3 t = E I.Pts + B.Pts + S.Pts 3 I.Pts + B.Pts + S.Pts 4 I.Pts + B.Pts + S.Pts t (c) 4 interior points, boundary points and source points with R = t (d) Comparison results of three cases with t = E-3 Figure 4- Results of vs. time in Example 3

15 U U (a) t =. (b) t = U U (c) t =. (d) t = 5. Figure 4-3 Results of wave motion vs. time in Example using 4 interior points, boundary points, source points with R = 4. and t = E-3 4

16 4.6e- 4.6e-.6e-.85e- 3.7e-.65e- 3.3e-.44e-.77e-.4e-.3e-.3e-.85e- 8.3e e e-3 9.4e-3 4.e e-3.93e e-3.88e-7 (a) t =., AME: 4.6 E- :.69 E- (b) t =., AME:.583 E- : E-3 7.4e e-3 3.6e-4 3.4e e-3.88e e-3.5e-4 4.e-3.6e-4 3.5e-3.8e-4.8e-3.44e-4 -.e-3 -.8e-4.4e-3 7.e e-4 3.e e-5.47e-8 (c) t =., AME: E-3 :.6348 E-3 (d) t = 5., AME: 3.67 E-4 : 8.84 E-5 Figure 4-4 Results of absolute error distribution vs. time in Example using 4 interior points, boundary points, source points with R = 4. and t = E-3 5

17 Example : the linear nonhomogeneous Klein-Gordon problem solved by the TM-MFS To solve Eq. (.) to (.4) with the Cassini domain and mixed boundary shown in Figure -, we can follow the way described in Example to generate the interpolation points distributing inside and outside the Cassini domain like the pattern of Figure 4-5 (left) and the test points shown in Figure 4-5 (right) for the error estimation of the approximate solution. In Example, we consider several cases with the different input data and list them in Table 4-. To implement the MFS+MPS, we choose, 3 and 4 quasi-random interior points inside the domain, respectively, Dirichlet boundary points and Neumann boundary points on the mixed boundary and source points around the circle with R = 3. and 3.5, respectively. The total,44 uniformly distributed test points are used to test the error of the approximate solution. Furthermore, the time interval including E-, 5E-3 and E-3 are used to discretize the time domain in the TM-MFS, respectively. As shown in Figure 4-6, the results of the versus time in each case have been obtained by the TM-MFS. From these results, it can be found that all s are the periodic values within E- to E-3 order error during the time history, and the s never keep ascending or descending when the time increases. In addition, the accurate approximate solution can also be obtained by using a small time interval (i.e. t = E-3). After observing the wave motion during the time history in Figure 4-7, we find that the wave repeatedly vibrates during the time history and it never decays. Therefore, the wave is a periodic and non-decaying motion and it can be completely represented by the exact solution of the linear nonhomogeneous Klein-Gordon equation, i.e. Eq. (.5). Due to the periodic wave motion, the trend of the results during the time history is the same as the wave motion results in Figure 4-7. Furthermore, we also estimate the results of the ARE and in the specific time using 4 interior points, boundary points, source points with R = 3.5 and t = E-3, and they are shown in Figure 4-8. For example, we obtain that the ARE is.4 E- and is.6733 E-3 at t =., and ARE is.649 E- and is.458 E-3 at t = 5. Table 4- Points and time interval for the TM-MFS in Example Case MPS MFS Interior points Boundary points Source points Source location a (D) + (N) R = 3. b 3 (D) + (N) R = 3. c 4 (D) + (N) R = 3.5 Houbolt FD t = E-, 5E-3 and E-3 Test points,84 uniformly distributed points and 6 boundary points 6

18 Figure 4-5 Point distribution (blue: interior points, green: Dirichlet boundary points, magenta: Neumann boundary points and red: source points) for the TM-MFS in Example (left) and test points (right) 7

19 t = E- t = 5E-3 t = E-3 t = E- t = 5E-3 t = E t t (a) interior points, boundary points and source points with R = 3. (b) 3 interior points, boundary points and source points with R = 3. t = E- t = 5E-3 t = E-3 - I.Pts + B.Pts + S.Pts 3 I.Pts + B.Pts + S.Pts 4 I.Pts + B.Pts + S.Pts t (c) 4 interior points, boundary points and source points with R = t (d) Comparison results of three cases with t = E-3 Figure 4-6 Results of vs. time in Example 8

20 U U (a) t =. (b) t = U - U (c) t = (d) t = 5 Figure 4-7 Results of wave motion vs. time in Example using 4 interior points, boundary points, source points with R = 3.5 and t = E-3 9

21 .5.e e-3 9.9e e-3 8.8e-3.96e e-3 6.6e e-3.e-3 5.5e-3.85e-3 4.4e-3.48e e e-3 -.e-3.e-3-7.4e-4 3.7e e e-6 (a) t =., max. ARE:.4 E- :.6733 E-3 (b) t = 5., max. ARE: E-3 : E-4.5.6e-.5.64e-.3e-.8e-.48e-.3e-.5.58e-.35e-.5.5e- 9.86e-3.3e- 8.e e e e e-3-4.5e-3.6e-3-3.9e-3.65e e e-6 (c) t =, max. ARE:.57 E- :.49 E-3 (d) t = 5, max. ARE:.649 E- :.458 E-3 Figure 4-8 Results of ARE distribution vs. time in Example using 4 interior points, boundary points, source points with R = 3.5 and t = E-3

22 Example 3: the linear homogenous Klein-Gordon problem solved by the TM-LMAPS We consider a large time domain problem which can be described by Eq. (.5) to (.7) and satisfies the boundary profile of the Amoeba shape domain with the mixed boundary condition, i.e. Eq. (.6) and (.7). To solve this problem by the TM-LMAPS, we can generate a large number of interpolation points distributing inside the Amoeba shape domain and on its boundary like the pattern of Figure 4-9 and use the same interpolation points as the test points for error estimation of the approximate solution. For the LMAPS, it is important to determine the number of the nearest neighbor points (n s ) around each center point for the interpolation in the local domain. Therefore, we have constructed several cases with the different input data and listed them in Table 4-3. In this example, we choose 3,496 uniformly distributed interior points inside the domain, 46 Dirichlet boundary points and 54 Neumann boundary points on the mixed boundary. Furthermore, the local nearest neighbor points, n s = 5,,, and the shape parameter of the MQ basis function, c = 5,, 3 are used for each cases, respectively. The total 3,896 interpolation points are also used to test the error of the approximate solution. The time interval including E-, 5E-3 and E-3 are used to discretize the time domain in the TM-LMAPS, respectively. As shown in Figure 4-, the results of the versus time in each case have been obtained by the TM-LMAPS. These results show that the keeps linearly descending when the time increases and the accurate approximate solution in each case can be obtained by using a small time interval (i.e. t = E-3). Checking the results in Figure 4- (d), it is clear to know that the acceptable approximate solution can be obtained when we only choose n s = 5 and t = E-3 since the s of three cases are very close. According to the wave motion results in Figure 4-, we can find that these results are very similar to the results in Figure 4-3. The initial transient wave will tend to be motionless after t = 5. Eq. (.9) completely represents this trend of the wave motion and the will approach to zero if the time is infinite. Furthermore, we also estimate the results of the AME and in the specific time using n s = 5 and t = E-3, and they are shown in Figure 4-. From this figure, we find that the worse AMEs almost occur on the Neumann boundary and no error occurs on the Dirichlet boundary. For example, we obtain that the AME is E- and is.4368 E-3 at t =., and AME is.88 E-4 and is.55 E-5 E-5 at t = 5.. Table 4-3 Points, shape parameter and time interval for the TM-LMAPS in Example 3 Case LMAPS Interior points Boundary points n s c a 3, (Dirichlet) + 54 (Neumann) 5 3 b 3, (Dirichlet) + 54 (Neumann) c 3, (Dirichlet) + 54 (Neumann) 5 Houbolt FD t = E-, 5E-3 and E-3 Test points 3,496 uniformly distributed points and 4 boundary points

23 Figure 4-9 Point distribution (blue: interior points, green: Dirichlet boundary points and magenta: Neumann boundary points) for the TM-LMAPS in Example 3

24 - t = E- t = 5E-3 t = E-3 - t = E- t = 5E-3 t = E t t (a) 3,496 interior points, 4 boundary points, ns = 5 and c = 3 (b) 3,496 interior points, 4 boundary points, ns = and c = - t = E- t = 5E-3 t = E ns = 5, c = 3 ns =, c = ns =, c = t t (c) 3,496 interior points, 4 boundary points, ns = and c = 5 (d) Comparison results of three cases with t = E-3 Figure 4- Results of vs. time in Example 3 3

25 U U (a) t =. (b) t = U U (c) t =. (d) t = 5. Figure 4- Results of wave motion vs. time in Example 3 using 3,496 interior points, 4 boundary points, ns = 5, c = 3 and t = E-3 4

26 (a) t =., AME: E- :.4368 E-3 (b) t =., AME:.547 E- :.949 E-3 (c) t =., AME: E-3 : 4.8 E-4 (d) t = 5., AME:.88 E-4 :.55 E-5 Figure 4- Results of absolute error distribution vs. time in Example 3 using 3,496 interior points, 4 boundary points, ns = 5, c = 3 and t = E-3 5

27 Example 4: the linear nonhomogeneous Klein-Gordon problem solved by the TM-LMAPS In this example, a large time domain problem is considered and described by Eq. (.) to (.4), and it satisfies the boundary profile of the Amoeba shape domain with the mixed boundary condition. For solving this problem by the TM-LMAPS, we use the interpolation points and test points which are the same as the point distribution of Example 3 (shown in Figure 4-9). Similarly, we also calculate the approximate solution and test the error from several cases listed in Table 4-3. Therefore, the linear nonhomogeneous Klein-Gordon problem, i.e. Eq. (.), is solved by using 3,496 uniformly distributed interior points, 4 boundary points, n s = 5,,, and c = 5,, 3, respectively. The time interval including E-, 5E-3 and E-3 are used to discretize the time domain in the TM-LMAPS, respectively. As shown in Figure 4-3, the results of the versus time in each case have been obtained by the TM-LMAPS. From these results, it can be found that all s are the periodic values within E- to E-4 order error during the time history, and the s never keep ascending or descending when time increases. In addition, the accurate approximate solution can also be obtained by using a small time interval (i.e. t = E-3). Checking the results in Figure 4-3 (d), we can find that the obtained by using n s = 5 and t = E-3 is less unstable than that obtained by using other parameters. Since using a few nearest neighbor points ( n s = 5) can provide a stable and acceptable approximate solution, we should try to choose the nearest neighbor points as few as possible for the interpolation in local domain when the TM-LMAPS is implemented for solving the time domain problem. From Figure 4-4, the wave motion is a periodic vibration during the time history and it never decays. This motion is similar to the result of Figure 4-7 in Example. Furthermore, we also estimate the results of the ARE and in the specific time using n s = 5 and t = E-3, and they are shown in Figure 4-5. This figure shows that the worse AMEs almost occur on the Neumann boundary and no error occurs on the Dirichlet boundary. For example, we obtain that the ARE is.96 E- and is E-4 at t =., and ARE is.87 E- and is E-4 at t = 5. 6

28 - t = E- t = 5E-3 t = E-3 - t = E- t = 5E-3 t = E t t (a) 3,496 interior points, 4 boundary points, ns = 5 and c = 3 (b) 3,496 interior points, 4 boundary points, ns = and c = - - t = E- t = 5E-3 t = E-3 - ns = 5, c = 3 ns =, c = ns =, c = t t (c) 3,496 interior points, 4 boundary points, ns = and c = 5 (d) Comparison results of three cases with t = E-3 Figure 4-3 Results of vs. time in Example 4 7

29 U U (a) t =. (b) t = 5. U - U (c) t = (d) t = 5 Figure 4-4 Results of wave motion vs. time in Example 4 using 3,496 interior points, 4 boundary points, ns = 5, c = 3 and t = E-3 8

30 (a) t =., max. ARE:.96 E- : E-4 (b) t = 5., max. ARE:.887 E- : E-4 (c) t =, max. ARE:.335 E- : E-4 (d) t = 5, max. ARE:.87 E- : E-4 Figure 4-5 Results of ARE distribution vs. time in Example 4 using 3,496 interior points, 4 boundary points, ns = 5, c = 3 and t = E-3 9

31 5. Discussions In Section 4, we have successfully solved four Klein-Gordon examples by using both global method (TM-MFS) and local method (TM-LMAPS). As shown in Table 5-, the results of the in the specific conditions obtained from each example have been listed in this table. According to these results, we obtain: () For obtaining the more accurate approximate solutions of Klein-Gordon equations, a small time interval (i.e. t = E-3) should be used for the Houbolt FD in both of methods. () Both of methods can obtain the quite accurate approximate solutions from each Klein-Gordon example, and all s of them are around E-3 to E-4 order error. (3) Comparing the results of all Klein-Gordon examples, it is found that the s obtained from the TM-LMPAS are superior to that obtained from the TM-MFS. This reason is that the dynamic results (e.g. wave motion) can be much clearly represented by using a large number of interpolation points. Since there is no ill-condition problem in the TM-LMAPS, the more accurate approximate solution can be obtained from the TM-LMAPS. (4) From the results of the linear nonhomogeneous Klein-Gordon problems (i.e. Examples and 4), the TM-MFS obtains around E-3 order and the TM-LMAPS obtains around E-4 order. (5) Figure 5- shows the results of the versus shaper parameter c of the MQ in Example 4. From these results, it is clear to know that the good solutions of Klein-Gordon problem can be obtained when we choose n s = 5 and c = 3 for the TM-LMAPS in Example 4. Furthermore, this figure also shows that n s = is still a good choices for the TM-LMAPS when c is less than 9. (6) In Example 4, the stable results of the versus time can be obtained by using a few nearest neighbor points (e.g. n s = 5) in the local domains. Table 5- Comparison results of in all Klein-Gordon examples Method Ex. Klein-Gordon problem type Domain Boundary Interpolation points Time..69 E- TM-MFS Linear homogenous Star Dirichlet 4~ E E E-5 (Global) Linear nonhomogen eous Cassini Dirichlet + Neumann 4~ E E-4.49 E E-3 TM-LMAPS 3 Linear homogenous Amoeba Dirichlet + Neumann 3, E E E E-5 (Local) 4 Linear nonhomogen eous Amoeba Dirichlet + Neumann 3, E E E E-4 3

32 ns = 5 ns = ns = ns = 5 ns = ns = c (a) 3,496 interior points, 4 boundary points, t = c (b) 3,496 interior points, 4 boundary points, t = 5. - ns = 5 ns = ns = - ns = 5 ns = ns = c (c) 3,496 interior points, 4 boundary points, t = c (d) 3,496 interior points, 4 boundary points, t = 5 Figure 5- Results of vs. shaper parameter c in Example 4 3

33 6. Conclusions In this report, we introduce both of meshless methods, the TM-MFS and TM-MAPS, to solve the Klein-Gordon equations. For numerical method verification, we have constructed four examples including the D linear homogenous and the -D nonhomogeneous Klein-Gordon problems and then solved them by using both TM-MFS and TM-MAPS, respectively. As a result, we are confident to show that our two proposed methods are quite useful for solving the Klein-Gordon equations. According to the results in Sections 4 and 5, some conclusions have been summarized as follows: () For solving the Klein-Gordon equations, we do not need to be an expert of the Klein-Gordon problems since we can easily solve them by our proposed methods. () Both TM-MFS and TM-LMAPS are useful for solving the time domain PDE problem, and which can obtain the quite accurate approximate solutions on Klein-Gordon problems. (3) All s of the approximate solutions obtained from the TM-MFS and TM-LMAPS are around E-3 to E-4 order error on Klein-Gordon problems. (4) For solving large and dynamic problems, we strongly recommend that the TM-LMAPS should be chosen as the numerical solver since it can provide the better results and save a lot of time on numerical interpolation. (5) To implement the TM-LMAPS, we recommend that the number of the nearest neighbor points, n s = 5, is a good choice to obtain the stable and good s on periodic problems and n s = is also good for the accurate results. 7. Reference [] Klein-Gordon equation, from Wikipedia, the free encyclopedia. [] D.L. oung, M.H. Gu, C.M. Fan, The time-marching method of fundamental solutions for wave equations, Engineering Analysis with Boundary Elements, p.4-45, 9. [3] M.H. Gu, C.M. Fan, D.L. oung, The method of fundamental solutions for the multi-dimensional wave equations, Journal of Marine Science and Technology, Vol. 9, No. 6, pp (). [4] C.S. Chen, Lecture Note, A localized MAPS. [5] The world of mathematical equation, Eqworld. [6] C.S. Chen, C.A. Brebbia, and H. Power, Dual reciprocity method using compactly supported radial basis functions, Commun. Numer. Meth. Engng, 5, 37-5 (999). 3