FTCS scheme
Imagine you have a toy car and you want it to move from one place to another. To move it, you have to push it with your hands. Now imagine you want the car to move continuously in a specific direction. You have to keep pushing it in the same direction every few seconds. This is similar to how the ftcs (Forward in Time, Central in Space) scheme works.
In mathematical terms, the ftcs scheme is a way to solve partial differential equations (PDEs) using a numerical method. PDEs are equations that describe the behavior of systems that change over time and space. For example, the temperature of a room will change over time and space, depending on factors such as the location of heating or cooling sources and the insulation of the walls.
To use the ftcs scheme, we divide the space we are interested in into small sections. Think of it like a grid on a map. We then evaluate the PDE at different points in the grid using the values of the variables at the neighboring points. This allows us to approximate the solution of the PDE at each point in the grid.
The "forward in time" part of the scheme means that we start at a specific time and move forward in small increments. At each time step, we use the current values of the variables and the values at neighboring points to calculate the values at the next time step.
The "central in space" part of the scheme means that we use the average value of the variables at the neighboring points to approximate the value at the central point. This helps to minimize error in the approximation.
Overall, the ftcs scheme is a way to numerically approximate solutions to partial differential equations. It breaks the problem down into smaller parts and uses information from neighboring points in space and time to estimate the solution at each point in the grid.
In mathematical terms, the ftcs scheme is a way to solve partial differential equations (PDEs) using a numerical method. PDEs are equations that describe the behavior of systems that change over time and space. For example, the temperature of a room will change over time and space, depending on factors such as the location of heating or cooling sources and the insulation of the walls.
To use the ftcs scheme, we divide the space we are interested in into small sections. Think of it like a grid on a map. We then evaluate the PDE at different points in the grid using the values of the variables at the neighboring points. This allows us to approximate the solution of the PDE at each point in the grid.
The "forward in time" part of the scheme means that we start at a specific time and move forward in small increments. At each time step, we use the current values of the variables and the values at neighboring points to calculate the values at the next time step.
The "central in space" part of the scheme means that we use the average value of the variables at the neighboring points to approximate the value at the central point. This helps to minimize error in the approximation.
Overall, the ftcs scheme is a way to numerically approximate solutions to partial differential equations. It breaks the problem down into smaller parts and uses information from neighboring points in space and time to estimate the solution at each point in the grid.