# Heat Equation Solver

The Heat Equation Solver is used for configuring the numerical solver settings for conjugate heat transfer simulations. It controls how the heat equation is solved within solid regions.

# 📋 Available Options

Option Description Unit
absolute_tolerance Absolute residual tolerance for heat equation convergence
equation_evaluation_frequency Frequency at which to solve the heat equation
order_of_accuracy Order of accuracy in space (fixed at 2)
linear_solver Settings for the linear solver

# 🔍 Detailed Descriptions

# absolute_tolerance

Absolute residual tolerance that determines the convergence of the heat equation in conjugate heat transfer.

  • Default: 1e-9
  • Example: 1e-10
  • Notes: This value should be the same or higher than the absolute tolerance for the linear solver by a small margin.

# equation_evaluation_frequency

Frequency at which to solve the heat equation in conjugate heat transfer simulations.

  • Default: 10
  • Example: 2
  • Notes: Lower values mean more frequent updates but higher computational cost.

# order_of_accuracy

Order of accuracy in space for the heat equation solver.

  • Default: 2
  • Example: 2
  • Notes: This value is fixed at 2 and cannot be changed.

# linear_solver

Configuration for the linear solver used to solve the heat equation.

  • Default: LinearSolver(max_iterations=50, absolute_tolerance=1e-10)
  • Example: See Python example below
  • Notes: Controls the inner linear solver iterations and convergence criteria.

💡 Tips
  • Set absolute_tolerance slightly higher than the linear solver's tolerance
  • Adjust equation_evaluation_frequency based on the coupling strength between fluid and solid regions
  • For steady simulations, higher evaluation frequencies can help with convergence
  • For unsteady simulations, lower frequencies might be needed to capture transient effects

❓ Frequently Asked Questions

  • How do I choose the right equation evaluation frequency?

    Start with the default value of 10. If the solution is not converging well, try decreasing it. For steady simulations with weak coupling, you can increase it to reduce computational cost.

  • Why can't I change the order of accuracy?

    The heat equation solver uses a fixed second-order accurate spatial discretization to ensure good accuracy while maintaining stability.

  • How should I set the absolute tolerance?

    Set it slightly higher than the linear solver's absolute tolerance to ensure proper convergence hierarchy. For example, if linear solver tolerance is 1e-10, use 1e-9 for the heat equation solver.

🐍 Python Example Usage

Below is a Python code example showing how to configure the Heat Equation Solver:

heat_solver = fl.HeatEquationSolver(
    equation_evaluation_frequency=2,
    absolute_tolerance=1e-9,
    linear_solver=fl.LinearSolver(
        max_iterations=50,
        absolute_tolerance=1e-10
    ),
    relative_tolerance=0.001
)

# Use it in a Solid model
solid_model = fl.Solid(
    entities=[volume_mesh["solid-*"]],
    heat_equation_solver=heat_solver,
    material=fl.SolidMaterial(...),
    initial_condition=fl.HeatEquationInitialCondition(temperature="1.0")
)

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