zbMATH — the first resource for mathematics

Solving partial differential equation by using multiquadric quasi-interpolation. (English) Zbl 1117.65134
Summary: We use a kind of univariate multiquadric (MQ) quasi-interpolation to solve a partial differential equation (PDE). We obtain the numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the temporal derivative of the dependent variable. The advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement. Our numerical experiment includes two examples. One is solving viscid Burgers’ equation for initial trapezoidal conditions. Another is simulating the interaction of two waves travelling in opposite direction. From the numerical experiment, we can see that the present scheme is valid.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
Full Text: DOI
[1] Beatson, R.K.; Dyn, N., Multiquadric B-splines, J. approx. theory, 87, 1-24, (1996) · Zbl 0864.41012
[2] Beatson, R.K.; Powell, M.J.D., Univariate multiquadric approximation: quasi-interpolation to scattered data, Constr. approx., 8, 275-288, (1992) · Zbl 0763.41012
[3] Chen, C.S.; Kuhn, G.; Li, J.C.; Mishuris, G., Radial basis functions for solving near singular Poisson problems, Commun. numer. meth. en., 19, 333-347, (2003) · Zbl 1018.65131
[4] Chen, R.H.; Wu, Z.M., Applied multiquadric quasi-interpolation to solve burgers’ equation, Appl. math. comput., 172, 472-484, (2006) · Zbl 1088.65086
[5] R.H. Chen, Z.M. Wu, Solving hyperbolic conservation laws using multiquadric quasi-interpolation, Numer. Meth. PDE, in press. · Zbl 1106.65087
[6] Fasshauer, G.E., Newton iteration with multiquadrics for the solution of nonlinear pdes, Comput. math. appl., 43, 423-438, (2002) · Zbl 0999.65136
[7] Fedoseyev, A.I.; Friedman, M.J.; Kansa, E.J., Continuation for nonlinear elliptic partial differential equations discretized by the multiquadric method, Int. J. bifurcat. chaos, 10, 481-492, (2000) · Zbl 1090.65550
[8] Friedman, M.J., Improved detection of bifurcations in large nonlinear system via the continuation of invariant subspace algorithm, Int. J. bifurcat. chaos, 11, 2277-2285, (2000) · Zbl 1091.65509
[9] Furzeland, R.M.; Verwer, J.G.; Zegeling, P.A., A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of line, J. comput. phys., 89, 349-388, (1990) · Zbl 0705.65066
[10] M.A. Golberg, C.S. Chen, Improved multiqudric approximation for partial differential equations, Eng. Anal. Bound. Elem. 18, 9-17.
[11] Hon, Y.C., Multiquadric collocation method with adaptive technique for problem with boundary layer, Int. J. appl. sci. comput., 6, 173-184, (1999)
[12] Hon, Y.C.; Lu, M.W.; Xue, W.M.; Zhu, Y.M., Multiquadric method for the numerical solution of a biphasic model, Appl. math. comput., 88, 153-175, (1997) · Zbl 0910.76059
[13] Hon, Y.C.; Mao, X.Z., A multiquadric interpolation method for solving initial value problems, J. sci. comput., 12, 51-55, (1997) · Zbl 0907.65062
[14] Hon, Y.C.; Mao, X.Z., An efficient numerical scheme for burgers’ equation, Appl. math. comput., 95, 37-50, (1998) · Zbl 0943.65101
[15] Hon, Y.C.; Wu, Z.M., A quasi-interpolation method for solving ordinary differential equations, Int. J. numer. meth. eng., 48, 1187-1197, (2000) · Zbl 0962.65060
[16] Hardy, R.L., Theory and applications of the multiquadric-biharmonic method, 20 years of discovery 1968-1988, Comput. math. appl., 19, 163-208, (1990) · Zbl 0692.65003
[17] Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics I, Comput. math. appl., 19, 127-145, (1990) · Zbl 0692.76003
[18] Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics II, Comput. math. appl., 19, 147-161, (1990) · Zbl 0850.76048
[19] Kansa, E.J.; Hon, Y.C., Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to ellipitic partial differential equations, Comput. math. appl., 39, 123-137, (2000) · Zbl 0955.65086
[20] Li, J.C.; Chen, C.S., Some observations on unsymmetric radial basis function collocation methods for convection – diffusion problems, Int. J. numer. meth. eng., 57, 1085-1094, (2003) · Zbl 1062.65504
[21] Sharan, M.; Kansa, E.J.; Gupta, S., Application of the multiquadric method for numerical solution of elliptic partial differential equations, Appl. math. comput., 84, 275-302, (1997) · Zbl 0883.65083
[22] Wong, A.S.M.; Hon, Y.C.; Li, T.S.; Chung, S.L.; Kansa, E.J., Multizone decomposition for simulation of time-dependent problem using the multiquadric scheme, Comput. math. appl., 37, 23-43, (1999) · Zbl 0951.76066
[23] Wu, Z.M., Solving differential equation with radial basis function, Advances computational mathematics, Lecture notes in pure and applied mathematics, vol. 202, (1999), Dekker, pp. 537-544
[24] Wu, Z.M., Dynamically knots setting in meshless method for solving time dependent propagations equation, Comput. meth. appl. mech. eng., 193, 1221-1229, (2004) · Zbl 1060.76633
[25] Wu, Z.M.; Schaback, R., Shape preserving properties and convergence of univariate multiquadric quasi-interpolation, ACTA math. appl. sinica, 10, 441-446, (1994) · Zbl 0822.41025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.