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A new numerical approach, called the “subdomain Chebyshev spectral method” is presented for calculation of the spatial derivatives in a curved coordinate system, which may be employed for numerical solutions of partial differential equations defined in a 2D or 3D geological model. The new approach refers to a “strong version” against the “weak version” of the subspace spectral method based on the variational principle or Galerkin’s weighting scheme. We incorporate local nonlinear transformations and global spline interpolations in a curved coordinate system and make the discrete grid exactly matches geometry of the model so that it is achieved to convert the global domain into subdomains and apply Chebyshev points to locally sampling physical quantities and globally computing the spatial derivatives. This new approach not only remains exponential convergence of the standard spectral method in subdomains, but also yields a sparse assembled matrix when applied for the global domain simulations. We conducted 2D and 3D synthetic experiments and compared accuracies of the numerical differentiations with traditional finite difference approaches. The results show that as the points of differentiation vector are larger than five, the subdomain Chebyshev spectral method significantly improve the accuracies of the finite difference approaches.

With development of computer technology, computer modeling that imitates the underground earth interior becomes now a powerful tool to geological, environmental and geophysical problems, e.g. numerical simulations of tectonic dynamics [

Spectral method, e.g. Chebyshev spectral method is superior to the traditional finite-difference approach in numerical differentiations because of exponential convergence [

In this paper, we present a subdomain Chebyshev spectral method to calculate spatial derivatives in a curved coordinate system, which is similar to subspace spectral element method (weak form) [

In general, computer modeling is to solve the following PDE [

subject to some Dirichlet or Neumann boundary conditions. Here

In order to match free-surface topography, e.g. hill, trench and ridge, one may employ the global coordinate transformations [

mapping the computational coordinates

Here, summation convention of the repeated integer subscripts k and l has been applied. Accordingly, the discrete versions of Equations (3) and (4) can be written in the following forms

where

Here,

Here,

reduced to the simplest forms:

spectral differentiation vectors [

tained, i.e.

Substituting Equations (5) and (6) into (1), and then applying the PDE to every point

which have 3N unknowns of

It should be noticed that the differentiation vectors

cable only if the Jacobian

ation (2) must be smooth (differentiable) in the domain. In the next section, we present a set of the local coordinate transformations which are applicable for arbitrary free-surface topography and make the numerical differentiation vectors

In order to exactly match free-surface topography and subsurface interfaces of the Earth, we divide the model domain

which maps the local coordinates

and both are functions of x and y. Therefore, Equation (11) gives non-zero components of the Jacobian matrix:

and non-zero components of the Hessian matrix:

Equations (13) and (14) indicate that the Jacobian and Hessian matrices depend on the derivatives

boundaries

spline interpolation functions to the boundaries

where

In the transformed subdomain

to three coordinates

and second differentiation vectors

where

Substitutions of Equations (13) and (17) for (7), we obtain general forms of the first differentiation vectors with respect to the global coordinates:

Similarly, substituting Equations (13), (14) and (19) into (8), we obtain the second differentiation vectors with respect to the global coordinates

and obtain

Here, we used

relationship between

It should be noticed that Equations (19) and (21) are valid for the points inside the subdomains. At a boundary point of the subdomains, they may yield different values of the partial derivatives due to multiple subdomains’ sharing. For the “strong” solution, the partial derivatives at the boundary points must be unique, for which we pick up half of Chebyshev points from the connecting subdomains (see points in the “boundary domain” in

To demonstrate the capability of the new approach, we designed a 2D and a 3D geological model (see

The former is a valley and the latter represents a ridge, both have two interfaces under the ground. With the given model geometries, we used subdomain Chebyshev points to discretise the model domain and obtained the subdomain Chebyshev differentiation grids.

In order to examine the accuracy of the subdomain Chebyshev spectral method, we employed the following testing functions as the physical quantities defined in the two models:

Here, L_{x}, L_{y}, and L_{z} are lengths of the model domain in three coordinate directions. From the two functions, one can analytically calculate the first and second partial derivatives, which are used to check the accuracies of the subdomain Chebyshev spectral method.

In order to compare the subdomain Chebyshev spectral method with the traditional finite-difference approach, we repeated the above numerical experiments with finite-difference method and compared the average and maximum absolutely relative errors of the two methods. Meanwhile, five algorithms have been implemented¾ three finite-difference approaches (FDM0, FDM1, FDM2) and two subdomain Chebyshev spectral methods (SCSM1, SCSM2). Here, the number “0” denotes the algorithm applying Lagrange interpolation functions to the free-surface topography

replacement of the global spline interpolations with local Lagrange interpolations, to match free-surface topography and subsurface interfaces, will lead numerical differentiation to fail in convergence (see FEM0-curves in

better convergences than the algorithms of FDM1 and SCSM1 in computations of the second derivatives, because the former does not require high-order derivatives

In the 3D case (see

from

We have demonstrated a subdomain Chebyshev spectral method to numerical differentiations in a curved coordinate system, which may be employed to seek a “strong” solution of PED defined in ageological or a geophysical model. The new method employs a set of non-linear local coordinate transformations, global spline interpolations and subdomain Chebyshev points to make the discretization grid automatically match free-surface topography and subsurface interfaces, and guarantees the numerical solution converging to the “strong” solution of partial differential equations. The new method possesses advantages from the traditional finite-difference approaches, i.e. simple discretization procedure and sparse assembled matrix, but it has stronger capability to deal with complex geometries of geological models and better accuracies in numerical differentiations. Methodologically speaking, the subdomain Chebyshev spectral method may replace the finite-difference approaches in geological or geophysical computer modelling and enables us to easily hand free-surface topography and significantly improve accuracy of modelling results.

2D and 3D synthetic experiments demonstrate the accuracies and capabilities of the subdomain Chebyshev spectral method. As points of the differentiation vectors are large than 5, the subdomain Chebyshev spectral method can reach one-order higher accuracy in numerical differentiations than the traditional finite-difference approaches in curved coordinates systems. Two algorithms of the subdomain Chebyshev spectral method (SCSM1, SCSM2) are both valid for numerical differentiations in PDE, but the algorithm SCSM2 is more efficient in computations than SCSM1, due to exclusion of computing the high-order derivatives of the coordinate trans- formations.

This work was supported by a Discovery Project (DP1093110) of the Australia Research Council. The authors thank Mr. Craig Patten for his assistance in using high-performance computing facility at eResearch SA in Australia.