#### 3.1. Trefftz Functions for Transient Moving Boundary Problems

The spacetime meshless method using Trefftz functions is rooted in the Trefftz method. Thus, it is necessary for the nonlinear moving boundary problems to formulate the general solutions. To formulate the transient Trefftz functions for the nonlinear moving boundary problems, the separation of variables is adopted.

where

$\phi (r,\theta )$ and

$\Omega (t)$ are functions. The total head

$h(r,\theta ,t)$ is a product of two functions. For simplicity, the following notations are considered.

where the subscript

r denotes the first derivative with respect to

r, the subscript

rr denotes the second derivative with respect to

r, the subscript

$\theta \theta $ denotes the second derivative with respect to

$\theta $. Inserting Equation (5) into Equation (1), by taking into account notation Equation (6), we have

We further consider the following equation

where

$R(r)$ and

$W(\theta )$ are functions. The function

$\phi (r,\theta )$ is a product of two functions, including

$R(r)$ and

$W(\theta )$. Each function depends only on one of the variables

r or

$\theta $. Inserting Equation (8) into Equation (7), we obtain

where

${R}^{\prime}=\frac{dR(r)}{dr}$,

${R}^{\u2033}=\frac{{d}^{2}R(r)}{d{r}^{2}}$,

${W}^{\u2033}=\frac{{d}^{2}W(\theta )}{d{\theta}^{2}}$, and

${\Omega}^{\prime}=\frac{d\Omega (t)}{dt}$.

Dividing by

$R(r)W(\theta )\Omega (t)$ on both sides in Equation (9), we can then obtain

where

$\lambda $ and

$\chi $ are separation constants. We introduce

p and

q and assume that

$\lambda ={p}^{2}$ or

$\lambda =-{p}^{2}$ and

$\chi ={q}^{2}$ or

$\chi =-{q}^{2}$ to ensure

$\lambda $ and

$\chi $ to be positive or negative value, respectively. The formulation of Trefftz functions for transient moving boundary problems are expressed in the following description. Considering the combination of positive or negative values for

$\lambda $ and

$\chi $, there are six possible scenarios. If we consider the first scenario,

$\lambda =0$ and

$\chi ={q}^{2}$, we may obtain

where

${A}_{1}$,

${A}_{2}$,

${A}_{3}$,

${A}_{4}$, and

${A}_{5}$ are arbitrary constants that have to be evaluated. Inserting Equation (11) into Equation (5) may yield

where

${\overline{A}}_{1}$,

${\overline{A}}_{2}$,

${\overline{A}}_{3}$, and

${\overline{A}}_{4}$ are arbitrary constants that have to be evaluated. We may find solutions for five other scenarios including

$\lambda =0$ and

$\chi =0$,

$\lambda ={p}^{2}$ and

$\chi ={q}^{2}$,

$\lambda ={p}^{2}$ and

$\chi =0$,

$\lambda =-{p}^{2}$ and

$\chi ={q}^{2}$, and

$\lambda =-{p}^{2}$ and

$\chi =0$ using the same procedure, as listed in

Appendix A. As a result, we may obtain the complete Trefftz functions described as follows,

where

**T** denotes the Trefftz basis functions, and

${\overline{\mathrm{T}}}_{1},{\overline{\mathrm{T}}}_{2},{\overline{\mathrm{T}}}_{3},\dots ,{\overline{\mathrm{T}}}_{18}$ denotes the functions, as listed in

Appendix A. The transient numerical solution for the two-dimensional subsurface flow problem with a transient moving boundary is expressed by the series expansion as follows,

where

$\upsilon $ denotes the order of the Trefftz functions, and

$\overline{a}$,

$\overline{b}$,

${\overline{c}}_{1q}$ …,

${\overline{d}}_{8qp}$ denote unknown coefficients,

${I}_{0}$ and

${I}_{q}$ denote the modified Bessel functions of the first kind of zero order and of

q order, respectively.

${J}_{0}$ and

${J}_{q}$ denote the Bessel functions of the first kind of zero order and of

q order, respectively.

${K}_{0}$ and

${K}_{q}$ denote the modified Bessel functions of the second kind of zero order and of

q order, respectively.

${Y}_{0}$ and

${Y}_{q}$ denote the Bessel functions of the second kind of zero order and of

q order, respectively.

For the infinite domain or domain with cavities, the Trefftz basis functions are described as

When the domain is simply connected, we may consider only positive basis functions. Consequently, the above equation is simplified as the following equation.

To evaluate the unknown coefficients of $\overline{a}$, ${\overline{c}}_{1q}$, …, ${\overline{d}}_{4qp}$ in Equation (16), the spacetime collocation scheme must be utilized. Using the spacetime collocation scheme and applying the Dirichlet boundary data in Equation (16), a system of equations may then be yielded.

where

${t}_{1},{t}_{2},\cdots ,{t}_{s}$ are time in dimensionless form,

${r}_{1},{r}_{2},\cdots ,{r}_{s}$ are radiuses in dimensionless form,

${\theta}_{1},{\theta}_{2},\cdots ,{\theta}_{s}$ are polar angles in dimensionless form,

${h}_{1},{h}_{2},\cdots ,{h}_{s}$ are Dirichlet boundary data, the subscript

s denotes the number of boundary points, and

$\overline{a},{\overline{c}}_{1q},\cdots ,{\overline{d}}_{4qp}$ denote the unknown coefficients. Equation (17) is expressed as

where

$\mathbf{H}$ denotes a matrix of the Trefftz basis functions with the size of

$s\times w$,

$\mathbf{y}$ denotes a vector of the unknown coefficients with the size of

$w\times 1$,

$\mathbf{Z}$ denotes a vector of accessible boundary value at boundary collocation points with the size of

$s\times 1$,

$s$ denotes the number of boundary points,

$w$ denotes the term related to the order of the Trefftz basis function. Solving Equation (18), we may acquire the coefficients that are unknown for the spacetime domain. In addition, the Neumann boundary conditions are also considered in this study.

where

$\stackrel{\rightharpoonup}{n}=({n}_{x},{n}_{y})$ denotes the outward normal vector,

${n}_{x}$ and

${n}_{y}$ are the outward normal direction of the

x and

y axis, respectively. Adopting the chain rule, we may yield the formulations of

${h}_{n}$,

${h}_{x}$, and

${h}_{y}$, as listed in

Appendix B.

#### 3.3. The Iterative Scheme for Modeling Transient Moving Boundary

For each collocation point on the moving surface, the total head is expressed as

where

${Y}_{j}$ is the height above the sea level,

$\gamma $ denotes the unit weight of water,

${p}_{j}$ denotes the pore water pressure,

${h}_{\phi}({r}_{j},{\theta}_{j},{t}_{j})$ is the total head, and the subscript

j denotes the index of the points on the transient moving boundary to be renewed. The over-specified moving boundary conditions, including the no-flux and the zero pressure head, are described as

For each collocation point on the seepage face, the total head is expressed as

As demonstrated in Equations (16) and (A11), the complete mathematical expressions of the Dirichlet and Neumann boundary conditions have been derived. Applying the Dirichlet and Neumann boundary values for boundary points on the moving surface may acquire

From the above equations, the given boundary data are over-specified on the moving boundary. On the moving boundary, the location of the moving boundary is unknown. It can be referred to as the inverse geometric problem. For example, considering the no-flux and the zero pressure head boundary conditions, the unknowns are the coordinates of collocation points. From Equations (23) and (24), it is found that we may solve a nonlinear system of equations to obtain the coordinates of collocation points for the given time. The moving boundary problem may, therefore, exhibit the nonlinear characteristic. The inverse geometric problems are usually difficult to deal with because of the nonlinearity. For solving the inverse geometric problem, such as the moving surface flow problem, the iterative scheme is required. Previous studies have found it difficult to calculate the Jacobian matrix using Newton’s method. Thus, the Picard iterative method is used in this study [

5]. The Picard iteration first begins from the initial guess of the location for the moving boundary. The iteration may be achieved by applying Equations (23) and (24).

where

${h}^{i}({r}_{j},{\theta}_{j},{t}_{j})={Y}_{j}^{i}$ and the superscripts

i is the number of iteration steps. The iterative equation is depicted as

where

${h}^{i}({r}_{j},{\theta}_{j},{t}_{j})$ is the total head to be renewed, and

$\epsilon $ is the parameter of under-relaxation. The

$\epsilon $ value is in the range of zero to one. The numerical procedure of the iteration starts by giving an initial value for the nonlinear moving boundary and ends while the stopping condition is achieved.

where

$\omega $ is the stopping criteria, and

J is the collocation point number on the moving boundary. In this study, we consider the stopping criteria to be

$\omega ={10}^{-4}$.