Group 1

Title

Koala*,1, Koala1, Koala1,P. H Gunawan1,2

Abstract
The aim of this research is devoted to study the simulation of two-dimensional heat equations using explicit scheme in parallel architecture. The physical problem of heat equations is illustrated in mathematical form. Some numerical simulations are given in order to show our parallel algorithm is implemented well for the numerical scheme.
CONTENTS
  1. Introduction ………………………………………………………
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1

  1. Mathematical formulation………………………………………..
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2

  1. Discrete Mathematics…………………………………………….
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3

  1. Parallel Architecture and Algorithm……………………………...
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4

  1. Numerical Simulations……………………………………………
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5

  1. Example 1…………………………………………………….
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4

  1. Example 2…………………………………………………….
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6

  1. Example 3……………………………………………………..
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7

  1. Conclusions
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7

  1. Introduction

In this section, the introduction of the group problem is introduced. The authors can add some information about the physical problem of the group problem. Parallel architecture also is introduced in detail [1-2]. Here, the goals of the chapter must be mentioned clearly and sequentially.

This paper is organized as follows, in Section 2, the mathematical formulation of physical problem is given. The discrete form of the mathematical formulation is elaborated in Chapter 3. Parallel architecture and algorithm is shown in Chapter 4. In Chapter 5, numerical simulations from various problems are demonstrated. Finally, the conclusions are drawn in Chapter 6.

  1. Mathematical formulation

In this chapter, the mathematical formulation is given. For instance, we give an overview of the physical problem.

Figure 1. The physical configuration of a wire with length.

We can add some figures, for example in Fig 1, with the caption size 11. And the equation for Fig 1, can be written as

where is the temperature, the diffusion coefficient, a Lipschitz function, and are space and time respectively.

The explanation of the equations (1-2) can be elaborated in detail in this section.

  1. Discrete mathematics

The discrete form of the equations (1-2) is given in this section. First the discrete domain of is constructed by divided the domain into partitions (see Fig. 2).

Figure 2. The partition of domain .

In the Fig. 2, we can see clearly that the domain is partitioned in partitions.

  1. Parallel architecture and algorithm

Following the theory in references [1-2], our programming structure will be constructed in parallel way. Two approaches in parallel architecture are given as distributed and shared parallel architecture. The algorithm of the scheme is shown in Algorithm 1.

Algorithm 1 The sequential of Fisher’s scheme
Step 0 / Start
Step 1 / Give the initial condition for .
Step 2 / Compute using (3.4) or (3.7) for explicit or semi-implicit scheme respectively.
Step 3 / Solve (2.1) for explicit scheme or (2.2) for semi-implicit scheme to get .
Step 4 / Update.
Step 5 / Repeat step 2 – step 4 until final time .
Step 6 / End
  1. Numerical simulations

Some numerical simulations will be presented in this section. First simulation is about the

5.1. Example 1

Here the first example will be elaborated in detail.Let the initial distribution density function. The domain is discretized with . The discrete time space is defined by (3.4). The evolution of density profile from time until is shown in Fig. 3.


Figure 3. The evolution of initial density profile in long time simulation.

We can see clearly that in Fig. 3, the population density converge to 1 in a long time simulation. This result is in a good agreement with the result obtained by Tveito, et. al., [3]: 344. Moreover, our result confirms that remains in unit interval, indicates a maximum principle.

5.2.Example 2

The second example is given as the initial distribution density function. The domain is discretized with.


Figure 4. The evolution of distribution initial density function . The left and right figures are shown the evolution of density for time and respectively.

Similar to the previous examples, this example also describes our solution converge to . The results can be seen in Fig. 4 on left or right side for the evolution of density for time or respectively . This also shows that the explicit scheme satisfies the minimum principle.

.

5.3. Example 3

Example 3 is here.

  1. Conclusions

The explicit numerical scheme for heat equation in parallel architecture is presented.

References

[1] Karniadakis, G. E., & Kirby II, R. M. (2003). Parallel scientific computing in C++ and MPI: a seamless approach to parallel algorithms and their implementation (Vol. 1). Cambridge University Press.

[2] Rauber, T., & Rünger, G. (2013). Parallel programming: For multicore and cluster systems. Springer Science & Business Media.

[3]Tveito, A., & Winther, R. (2005). Introduction to partial differential equations: a computational approach (Vol. 29). Springer Science & Business Media.

About the authors

Koala
Student ID:123448999
Research interest: Parallel for Scientific computing / Koala
Student ID:123448999
Research interest: Parallel for Scientific computing
Koala
Student ID:123448999
Research interest: Parallel for Scientific computing / Dr. Putu Harry Gunawan
Lecture ID:15865761-6
Research interest: Numerical Mathematics, CFD, Parallel for Scientific Computing

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