Adaptive Fup collocation method for time dependent partial differential equations.

Gotovac, Hrvoje ; Kozulic, Vedrana ; Gotovac, Blaz 等

1. INTRODUCTION

Classical solving of the partial differential equations or boundary-initial value problems is reduced to the time integration of the ordinary differential equations in respect to the spatial discretization and corresponding boundary conditions. This approach enforces the same time step for all spatial locations, which is not optimal for problems which are simultaneously intermittent in both space and time. The usual time marching schemes do not provide any control over the global error in time.

Another approach means the solving of the PDE's simultaneously in the space-time domain. Significant improvements have been obtained by the adaptive space-time finite elements. Recently, there have been many attempts to develop new adaptive procedures which, among others, are focused upon using the adaptive wavelet collocation methods (Alam et al., 2006). All previous existing algorithms with wavelets and splines used localized basis functions only to obtain efficient adaptive strategy, but the PDE is solved by finite difference method on non-uniform adaptive grid (including all levels).

In this work, infinitely differentiable functions with compact support are used. These functions, called Fup basis functions, are one type of Rvachev's or atomic basis functions. Recently, Gotovac et al. (Gotovac et al., 2007; Kozulic et al., 2007) established Adaptive Fup Collocation Method (AFCM) with adaptive spatial algorithm, but classical time marching algorithm. In this paper, we present novel form of the AFCM with only Fup basis functions at each level using the collocation framework in space-time domain. Spatial discretization and grid adaptation depend on the character of the solution and accuracy criteria. An efficiency of the proposed meshless method is illustrated by numerical solving of Burgers equation.

2. Fup BASIS FUNCTIONS

The [Fup.sub.n](x) function support is determined according to:

supp [Fu.sub.n](x) = [-(n + 2) [2.sup.-n-1]; (n + 2) [2.sup.-n-1] (1)

Index n denotes the highest degree of the polynomial which can be expressed accurately in the form of a linear combination of [Fup.sub.n] (x) functions displaced by a characteristic interval [2.sup.-n]. Procedures for calculation of Fup function values are given in ref. (Gotovac & Kozulic, 1999) together with an illustration of their properties. Figure 1 shows the [Fup.sub.4](x) function and its first two derivatives.

[FIGURE 1 OMITTED]

3. ADAPTIVE FUP COLLOCATION METHOD FOR BOUNDARY-INITIAL VALUE PROBLEM

Generally, one-dimensional boundary-initial value problem can be described by the following nonlinear time-dependent partial differential equation:

LI u (x,t) [equivalent to] [partial derivative] u(x,t)/[partial derivative]t + KI u(x,t) = f (x,t), x [member of] ([X.sub.t], [X.sub.2]), t [member of] (0, T) (2)

and corresponding boundary and initial conditions:

LBu ([X.sub.b], t) = [g.sub.b] ([X.sub.b], t), b = 1,2, t [member of] (0,T) (3)

u(x,0) = [u.sub.0] (x), x [member of] ([X.sub.1], [X.sub.2]), t = 0 (4)

where u( x,t ) is a function that depends on one spatial variable x, LI and LB are partial and boundary differential operator, respectively, KI is an operator that consists of partial derivatives with respect to x only, while f, [g.sub.b] and [u.sub.0] are known functions.

We can consider this problem as a boundary value problem in the space-time domain. Basis function for numerical analyses of 2D problems is obtained from the Cartesian product of two one-dimensional Fup basis functions defined for each direction.

Fup collocation discretization reduces the problem to a system of algebraic equations for domain collocation points:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

for the boundary collocation points:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

and for the boundary that refer to the initial time:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

In Eqns. (5)-(7) J shows level from zero to maximum level needed for a desired accuracy, [d.sub.j.sub.kI] are Fup coefficients, [[phi].sup.j.sub.k,l] are Fup basis functions, k presents the index of collocation points at the current level for x-direction, l presents the index of collocation points at the current level for t-direction, [J.sub.minx] and [J.sub.mint] are numbers of collocation points at zero level in x and t directions, respectively, and [r.sup.J] is the residual vector. For the boundary that refer to the final time, PDE is satisfied. System (5)-(7) satisfies the differential flow equation in the internal collocation points (internal Fup coefficients) and boundary conditions in the corresponding boundary collocation points (external Fup coefficients).

The given differential equation is solved only at the zero level of resolution. Each non-zero level solves only residual of the solution from all previous levels and gives particular solution correction. Adaptive criterion adds new collocation points in the next level only in the zones where solution correction is greater than the prescribed threshold.

Numerical solving of the problem is performed by use Newton's method. The method starts with a suitable initial space-time grid and an initial guess for the solution of the problem equation which incorporates the Dirichlet boundary values. The entire space-time mesh is solved at once and can be used by an error estimator to iteratively compute a new adapted space-time mesh.

4. NUMERICAL EXAMPLE

Burgers equation is resulted from the application of the Navier--Stokes equation to unidirectional flow without pressure gradient and small viscosity. Problem is described by the following equation, initial and boundary conditions:

[partial derivative]u/[partial derivative]t = v [[partial derivative].sup.2]u/[partial derivative][x.sup.2] - u [partial derivative]u/[partial derivative]x (8)

u(x,0) = -sin([[pi] x) (9)

u([+ or -] 1, t) = 0 (10)

where u is the dimensionless velocity, while domain and viscosity are defined by: x [member of] [-1, 1] ; v = [10.sup.-2]/[pi].

Solution is characterized with one dimensional shock that is fixed in space, but rapidly increases in time. Shock is very narrow due to small viscosity. Fig. 2 presents numerical solution in x-t domain obtained with space-time AFCM using [Fup.sub.4] (x,t) basis functions. We were considered the final time T = 0.4 which is sufficient for smooth initial condition to become highly intermittent. Initial grid is determined by [j.sub.min x] = 8 and [j.sub.min t] = 2. Grid adaptation is performed both in space and time directions as shown in Fig. 3. Local time step is particularly interested. The time scale is fastest where the gradient steepens.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

5. CONCLUSION

The proposed meshless method solves nonlinear PDE's simultaneously in the space-time domain. Numerical algorithm implements infinitely differentiable basis functions with compact support and the collocation technique. Grid is adapted progressively by setting the threshold as direct measure of the solution correction at some level. In contrast to the classical time stepping schemes where globally accumulated error can arise and is not easy to adapt to multiple time stepping, the space-time AFCM provides all space and time multiple scales and the global error is controlled in time by a priori threshold.

6. REFERENCES

Alam, J.M.; Kevlahan, N.K.R. & Vasilyev, O.V. (2006). Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations. J. Comput. Phys., Vol. 214 (2006), pp 829-857

Gotovac, B. & Kozulic, V. (1999). On a selection of basis functions in numerical analyses of engineering problems. Int. J. Engineering Modelling, Vol. 12, No. 1-4 (1999), pp 25-41

Gotovac, H.; Andricevic, R. & Gotovac, B. (2007). Multi-resolution adaptive modeling of groundwater flow and transport problems. Adv. in Water Resour., Vol. 30, No. 5 (2007), pp 1105-1126

Kozulic, V.; Gotovac, H. & Gotovac, B. (2007). An Adaptive Multi-resolution Method for Solving PDE' s. Computers, Materials, and Continua, Vol. 6, No. 2 (2007), pp 51-70