Analytic-Numerical Study for (1+1) Cauchy Problem of Mixed Hyperbolic-Elliptic Type

Department of Mathematics and Statistics, Mutah University, Mutah P.O.Box 7, Al-Karak – Jordan
[email protected]

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Abstract. This work provides a technical description of the application of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic-elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, that raise in diverse branches of physics and engineering, reveal the reliability, efficiency and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics.

Keywords: Residual power series method; hyperbolic-elliptic system; conservation laws; Cauchy problem; Error analysis; Numerical simulation; Stability; Riemann problem.
AMS subject classifications: 35B35; 35L65; 65M12; 65M15.
1. INTRODUCTION
A first order partial differential equation of the form
(1)
is known as a scalar conservation law in one space dimension. is the partial differential transform with respect to , represents the conserved quantity, is the flux, is the one-dimensional space variable, while denotes the time. This type of equations often describes transport phenomena.
In physics, laws of conservation based on several principles state that certain measurable quantities do not change during time within an isolated physical system. i.e. the quantity is neither created nor destroyed. In classical physics and mechanics, they govern energy, momentum, angular momentum, mass, and electric charge. In particle physics, other conservation laws apply to properties of subatomic particles that are invariant during interactions.
Recall that system of conservation laws is hyperbolic at a point on a given domain if the Jacobian matrix of the flux vector has real eigenvalues. Moreover, if these eigenvalues are distinct, the system is strictly hyperbolic. In the case of complex conjugate eigenvalues, the system is elliptic at that point. Elliptic region denotes the set of all points at which the system is elliptic. Systems with nonempty two regions, hyperbolic and elliptic are called mixed systems.
The generic structure of solving mixed systems is not yet fully understood as in the case of hyperbolic systems 1. Mathematical ill-posedness of the system in the elliptic region reveals the physical fact that the state in the elliptic region is not stable, and it typically evolves into phase transitions.
With unknown exact solutions for many physical phenomena governed by systems of conservation laws, the obtained approximate solutions allow physicists to draw conclusions in an efficient way. For mathematics, such studies make the pure mathematics more meaningful. Because of this foremost importance, many authors have devoted their attention to study systems of conservation laws. For overviews of the mathematical theory for this class of problems and their applications, see 2-7 and references therein. A collection of analytic, semi-analytic and numerical attempts on solving mixed systems have been carried out in 8-14.
This work interests in constructing solutions for the mixed couple of scalar conservation laws Eq.(1), known as general Cauchy problems, via recently developed technique, named by the residual power series method (RPSM). Such systems arise as a model for a diverse range of physical phenomena from traffic flow to three-phase flow in porous media 11. Three examples with different properties are discussed. Comparing with some other used techniques shows that the proposed scheme is powerful and reliable.
2. METHODOLOGY AND ERROR ANALYSIS
The RPSM was first developed as an efficient method for fuzzy differential equations 15. A wide range of applications to fractional partial differential equations have been achieved 16-21.
Consider the generalized 1D scalar system of conservation laws
, , (2)
where and are assumed to be analytic on . In more compact form, and using the vector symbols, system in Eq.(2) can be written as
. (3)
In current section, we illustrate the methodology of the residual power series technique for treating the system in Eq.(3). The RPSM assumes the solution in a form of power series
, (4)
with .
For , define the -order approximate solution, which satisfies initial conditions, by the truncated series
, (5)
The first approximate solution, with in Eq.(4), of would be
,
subject to:
,
where,
(6)
is the analytic -residual vector function. Continuing this process for , the unknown coefficients could be determined consecutively by solving the system of algebraic equations
. (7)
Accordingly, the following facts for the RPSM were proved 22:
Theorem 2.1. Suppose that the functions has the power series representation at in Eq.(4) on the domain , for some closed interval , is the radius of convergence for the power series. If , then the unknown coefficients, for , are given by
. (8)
If is analytic on the indicated domain, general solution will be as in Eq.(4), provided that the series have closed forms.
Theorem 2.2. The residual function vanishes as approaches the infinity.
Corollary 2.3. The truncated series solution given in Eq.(5) and obtained by the RPSM is the – order Taylor expansion of about .
As a result of Corollary 2.3, and because of convenience to have a notation on error analysis for the approximate solution, Taylor’s theorem tells that
,
where
,
for some , is the reminder term vector. Assume that , positive constants , exist with , and in the sense of norm, , we get the following result:
Corollary 2.4. Suppose that . If the truncated series solution given in Eq.(5) is used to approximate the solution of the initial-value problem Eq.(3), then error bound is estimated to be
. (9)
The error bound for is something large since the values of are of distant from the center. So, constructing multistep-RPSM, as we did in the case of Adomian-Rach decomposition method 23, will reduce the error bound according to required tolerance. For the numerical accuracy and comparison purposes, absolute and relative errors, defined at each point , are given by
, (10)
and
, (11)
respectively. This is useful with given closed-form exact solution. In the case of unknown exact solution, absolute errors will be defined in two ways as following:
, (12)
or, by direct substitution into Eq.(3),
, (13)
While, the relative error is defined by
. (14)
Eq.(10), and therefore Eq.(13), implies that the truncated series solution converges to exact solution , if as . That means, the series solution approaches the exact solution if the sequence of real number approaches the unity as , where
, (15)
is the -convergence rate. Faster convergence can be obtained if with small number of iterations.
3. THE CAUCHY PROBLEM – NUMERICAL ILLUSTRATIONS
This section aims to verify the convergence and stability of the proposed algorithm to couple of mixed-type scalar conservation laws Eq.(1). The mixed-type Cauchy-problem with flux vector
, (16)
subject to two types of continuous initial data is considered. The elliptic region can be determined easily to be
. (17)
A prototype of mixed-type Cauchy problem with Riemann initial data is treated in Example 3.3.
Example 3.1. Consider the separated initial-valued Cauchy problem, Eq.(3) and Eq.(16), subject to:
. (18)
Operating the residual power series algorithm, coefficients of the series solution Eq.(4), that derived in Eq.(7), are listed to be:
,
,
,

.
Applying the ratio test, the series converges for and . Continuing this process, we get the exact solution
,
obtained upon using the Taylor’s expansion. An error bound for using the th order approximate solution can be determined using Eq.(9). In comparison to obtained results using the Adomian decomposition method (ADM) 8 and the variational iteration method (VIM) 14, the RPSM is more accurate on a long domain with clear differences.
The solution approximately enters the elliptic region at and some finite space as exhibited in Figure 1. The Glimm’s 24 and finite difference 5 schemes employed to treat this problem numerically. It is clear that the both are of high instability while transition through regions. Figure 2 shows the numerical solution using our proposed scheme. The solution keeps stability while the system changes type.

Fig.1. Elliptic region Eq.(19) in the – plane for Example 3.1.
Example 3.2. With nonlinear periodic initial conditions:
, (19)

Fig.2. Parametric-numerical solution using the RPSM in Example 3.1, for , and step-size .
the RPSM is implemented to solve the Cauchy problem considered in the previous example. While more coefficients of series solution Eq.(4) could be determined easily with aid of Mathematica software package, the approximate truncated-series solution Eq.(5), of order , will be used in coming numeric-analytic analysis. The first three terms of truncated series solution are:
,

Both sequences of coefficients are neither arithmetic nor geometric. So, the problem doesn’t have closed-form exact solution. The approximate solutions are shown in Figure 3. To obtain the high accuracy of our scheme, the corresponding absolute errors according to Eq.(13) are calculated and graphed, for , in Figure 4. Rates of convergence Eq.(15) are listed to be:

Fig.3. 3D-surface plots of 10th-order approximate solutions of (a) and (b) using the RPSM in Example 3.2, for , .

Fig.4. 3D-surface plots of absolute errors (a) and (b) using the RPSM in Example 3.2, for , .
As a numerical illustration of instability of the finite difference method, this example is suggested by Holden et al 5 with initials start outside the elliptic region (Figure 5), then enters this region for some finite time and space . The obtained solutions, using the RPSM, are stable and the transition stability of numerical solution in the – plane is clearly visible in Figure 6.

Fig.5. Elliptic region Eq.(19) in the – plane for Example 3.2.

Fig.6. Parametric-numerical solution using the RPSM in Example 3.2, for , and step-size .
Example 3.3. Consider a prototype of mixed-type Cauchy problem Eq.(3), that modeling three phase flow in porous media, with the flux function
. (20)
The eigenvalues of the associated Jacobian matrix are given by
,
such that the system being hyperbolic for
,

and elliptic for
. (21)
With Riemann initial-data
, (22)
where , and , this problem is considered in 1 to examine the development of oscillations over a small number of time steps numerically.
The RPSM is designed to deal with continuous initial data, so the following approximate transformation 12, 13
, (23)
will convert the problem into continuous initial-valued problem, and the residual power series algorithm could be implemented. Here is suitability constant. Consequently, transformed initial conditions will be
. (24)
The assumed power series solution Eq.(4) will be improved to be
, (25)
Following the residual power series procedure subject to the initial data
, (26)
the 10th-order approximate solution is obtained with aid of Mathematica. Several profiles of this solution are shown in Figure 7. Applying the formula in Eq.(13), Figure 8 shows the associated absolute errors for . An error bound on the interval can be determined by combining Eq.(9) and Eq.(12), with and , to be
. (27)

Fig.7. 2D-profiles of the 10th-order approximate solutions of (a) and (b) using the RPSM in Example 3.3, for and .

Fig.8. 3D-surface plots of absolute errors (a) and (b) using the RPSM in Example 3.3, for , .
In comparison to the ADM and the VIM 26, the efficiency, with higher order solution, and simplicity, with no need for complex integrals computing, of the RPSM for treating mixed-type problems is observed. The rates of convergence are as following:

The initials start from inside the elliptic region Eq.(21), then leave to hyperbolic as shown in Figure 9. The solutions still stable with no oscillations as shown in Figure 7 and confirmed in Figure 10.

Fig.9. Elliptic region Eq.(21) in the – plane for Example 3.3.

Fig.10. Parametric-numerical solution using the RPSM in Example 3.3, for , and step-size .
4. CONCLUSIONS
The study outlines the significant features of the residual power series method in constructing semi-analytic solutions for hyperbolic-elliptic systems of conservation laws. The obtained solutions are expressed in a form of convergent power series with easily computable components. Convergence and error analysis are included. The scheme is numerically tested on the general Cauchy problem of mixed type. It is modified to tackle Riemann initial-valued Cauchy problems. The stability of approximate solutions is obtained graphically when system changes type. Comparing to other existing methods, the results show the efficiency, convenience and applicability of the residual power series algorithm for a wide class of problems in science.
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