#### Spherical Blast Wave Simulation

Posted 2nd December 2020 by Holger Schmitz

Here is another animation that I made using the Vellamo fluid code. It shows two very simple simulations of spherical blast waves. The first simulation has been carried out in two dimensions. The second one shows a very similar setup but in three dimensions.

You might have seen some youtube videos on blast waves and dimensional analysis on the Sixty Symbols channel or on Numberphile. The criterion for the dimensional analysis given in those videos is true for strong blast waves. The simulation that I carried out, looks at the later stage of these waves when the energy peters out and the strong shock is replaced by a compression wave that travels at a constant velocity. You can still see some of the self-similar behaviour of the Sedov-Taylor solution during the very early stages of the explosion. But after the speed of the shock has slowed down to the sound speed, the compression wave continues to travel at the speed of sound, gradually losing its energy.

The video shows the energy density over time. The energy density includes the thermal energy as well as the kinetic energy of the gas.

For those of you who are interested in the maths and the physics, the code simulates the Euler equations of a compressible fluid. These equations are a model for an ideal adiabatic gas. For more information about the Euler equations check out my previous post.

#### Computational Physics Basics: Integers in C++, Python, and JavaScript

Posted 5th August 2020 by Holger Schmitz

In a previous post, I wrote about the way that the computer stores and processes integers. This description referred to the basic architecture of the processor. In this post, I want to talk about how different programming languages present integers to the developer. Programming languages add a layer of abstraction and in different languages that abstraction may be less or more pronounced. The languages I will be considering here are C++, Python, and JavaScript.

### Integers in C++

C++ is a language that is very close to the machine architecture compared to other, more modern languages. The data that C++ operates on is stored in the machine’s memory and C++ has direct access to this memory. This means that the C++ integer types are exact representations of the integer types determined by the processor architecture.

The following integer datatypes exist in C++

Type | Alternative Names | Number of Bits | G++ on Intel 64 bit (default) |
---|---|---|---|

`char` |
at least 8 | 8 | |

`short int` |
`short` |
at least 16 | 16 |

`int` |
at least 16 | 32 | |

`long int` |
`long` |
at least 32 | 64 |

`long long int` |
`long long` |
at least 64 | 64 |

This table does not give the exact size of the datatypes because the C++ standard does not specify the sizes but only lower limits. It is also required that the larger types must not use fewer bits than the smaller types. The exact number of bits used is up to the compiler and may also be changed by compiler options. To find out more about the regular integer types you can look at this reference page.

The reason for not specifying exact sizes for datatypes is the fact that C++ code will be compiled down to machine code. If you compile your code on a 16 bit processor the plain `int`

type will naturally be limited to 16 bits. On a 64 bit processor on the other hand, it would not make sense to have this limitation.

Each of these datatypes is signed by default. It is possible to add the `signed`

qualifier before the type name to make it clear that a signed type is being used. The `unsigned`

qualifier creates an unsigned variant of any of the types. Here are some examples of variable declarations.

char c; // typically 8 bit unsigned int i = 42; // an unsigned integer initialised to 42 signed long l; // the same as "long l" or "long int l"

As stated above, the C++ standard does not specify the exact size of the integer types. This can cause bugs when developing code that should be run on different architectures or compiled with different compilers. To overcome these problems, the C++ standard library defines a number of integer types that have a guaranteed size. The table below gives an overview of these types.

Signed Type | Unsigned Type | Number of Bits |
---|---|---|

`int8_t` |
`uint8_t` |
8 |

`int16_t` |
`uint16_t` |
16 |

`int32_t` |
`uint32_t` |
32 |

`int64_t` |
`uint64_t` |
64 |

More details on these and similar types can be found here.

The code below prints a 64 bit `int64_t`

using the binary notation. As the name suggests, the `bitset`

class interprets the memory of the data passed to it as a bitset. The bitset can be written into an output stream and will show up as binary data.

#include <bitset> void printBinaryLong(int64_t num) { std::cout << std::bitset<64>(num) << std::endl; }

### Integers in Python

Unlike C++, Python hides the underlying architecture of the machine. In order to discuss integers in Python, we first have to make clear which version of Python we are talking about. Python 2 and Python 3 handle integers in a different way. The Python interpreter itself is written in C which can be regarded in many ways as a subset of C++. In Python 2, the integer type was a direct reflection of the `long int`

type in C. This meant that integers could be either 32 or 64 bit, depending on which machine a program was running on.

This machine dependence was considered bad design and was replaced be a more machine independent datatype in Python 3. Python 3 integers are quite complex data structures that allow storage of arbitrary size numbers but also contain optimizations for smaller numbers.

It is not strictly necessary to understand how Python 3 integers are stored internally to work with Python but in some cases it can be useful to have knowledge about the underlying complexities that are involved. For a small range of integers, ranging from -5 to 256, integer objects are pre-allocated. This means that, an assignment such as

n = 25

will not create the number 25 in memory. Instead, the variable `n`

is made to reference a pre-allocated piece of memory that already contained the number 25. Consider now a statement that might appear at some other place in the program.

a = 12 b = a + 13

The value of `b`

is clearly 25 but this number is not stored separately. After these lines `b`

will reference the exact same memory address that `n`

was referencing earlier. For numbers outside this range, Python 3 will allocate memory for each integer variable separately.

Larger integers are stored in arbitrary length arrays of the C `int`

type. This type can be either 16 or 32 bits long but Python only uses either 15 or 30 bits of each of these "digits". In the following, 32 bit `int`

s are assumed but everything can be easily translated to 16 bit.

Numbers between −(2^{30} − 1) and 2^{30} − 1 are stored in a single `int`

. Negative numbers are not stored as two’s complement. Instead the sign of the number is stored separately. All mathematical operations on numbers in this range can be carried out in the same way as on regular machine integers. For larger numbers, multiple 30 bit digits are needed. Mathamatical operations on these large integers operate digit by digit. In this case, the unused bits in each digit come in handy as carry values.

### Integers in JavaScript

Compared to most other high level languages JavaScript stands out in how it deals with integers. At a low level, JavaScript does not store integers at all. Instead, it stores all numbers in floating point format. I will discuss the details of the floating point format in a future post. When using a number in an integer context, JavaScript allows exact integer representation of a number up to 53 bit integer. Any integer larger than 53 bits will suffer from rounding errors because of its internal representation.

const a = 25; const b = a / 2;

In this example, `a`

will have a value of 25. Unlike C++, JavaScript does not perform integer divisions. This means the value stored in `b`

will be 12.5.

JavaScript allows bitwise operations only on 32 bit integers. When a bitwise operation is performed on a number JavaScript first converts the floating point number to a 32 bit signed integer using two’s complement. The result of the operation is subsequently converted back to a floating point format before being stored.