Advanced Method of Characteristics Solver Calculator

Calculator Inputs

Enter expressions with variables x, y, u, r, and s. Allowed functions include sin, cos, tan, exp, log, sqrt, abs, pow, min, and max.

a(x, y, u)

Coefficient for dx/ds.

b(x, y, u)

Coefficient for dy/ds.

c(x, y, u)

Source term for du/ds.

Initial x₀(r)

Starting x on the initial curve.

Initial y₀(r)

Starting y on the initial curve.

Initial u₀(r)

Initial state along the curve.

r minimum

r maximum

Number of characteristics

Maximum march distance s

Step size

Numerical method

Target x

Target y

Example preset

Default inputs solve u_x + 2u_y = 0 with y = 0 and u(r, 0) = sin(r).

Reset Form

Example Data Table

Example based on the default setup: a = 1, b = 2, c = 0, x₀(r) = r, y₀(r) = 0, u₀(r) = sin(r).

r	s	x = r + s	y = 2s	u = sin(r)	Interpretation
-1.0000	1.0000	0.0000	2.0000	-0.8415	Characteristic starting left of the origin.
0.0000	1.0000	1.0000	2.0000	0.0000	This path lands exactly on the sample target.
1.0000	1.0000	2.0000	2.0000	0.8415	Characteristic starting right of the origin.
0.5000	0.5000	1.0000	1.0000	0.4794	Intermediate point before reaching larger y values.

Formula Used

The calculator handles a first order PDE written as:

a(x, y, u)u_x + b(x, y, u)u_y = c(x, y, u)

It then traces the characteristic system:

dx/ds = a(x, y, u), dy/ds = b(x, y, u), du/ds = c(x, y, u)

Initial data is supplied through:

x(0) = x₀(r), y(0) = y₀(r), u(0) = u₀(r)

The page numerically advances each characteristic using Euler or fourth order Runge Kutta. It estimates the requested solution by selecting the traced point nearest to the target coordinate.

How to Use This Calculator

Enter the PDE coefficients a(x, y, u), b(x, y, u), and c(x, y, u).
Define the initial curve with x₀(r), y₀(r), and u₀(r).
Choose the parameter range and number of characteristics to trace.
Set the maximum marching distance and the integration step size.
Enter the target coordinate where you want the estimated solution.
Select RK4 for better accuracy or Euler for quick rough checks.
Press Solve Characteristics to show the result above the form.
Use the CSV and PDF buttons to export the traced output.

Frequently Asked Questions

1) What type of PDE does this calculator support?

It supports first order equations written in characteristic form with two independent variables and one dependent variable. The coefficients may depend on x, y, and u.

2) Is the reported value exact?

Not always. The page computes a numerical estimate from traced characteristics. Accuracy improves when the grid of characteristics is dense and the step size is small.

3) What do r and s represent?

The parameter r labels starting points on the initial curve. The variable s is the marching parameter used to advance each characteristic path through the domain.

4) Should I choose RK4 or Euler?

RK4 is usually the better default because it is more accurate for smooth problems. Euler is useful for fast testing, but it may need much smaller steps.

5) Why does the target point sometimes not match exactly?

The tool traces discrete points rather than a continuous surface. If no traced point lands on the target, it returns the nearest one and reports the distance.

6) How can I improve numerical accuracy?

Reduce the step size, increase the number of characteristics, and extend the marching range when needed. These changes improve resolution around the target point.

7) What happens if characteristics cross?

Crossing characteristics can signal loss of a single valued smooth solution. The calculator still traces paths, but interpretation becomes problem dependent and should be checked carefully.

8) Can the source term depend on the solution?

Yes. You may let c depend on x, y, and u. The same is true for a and b, which makes the traced system fully state dependent.