Take this code: --[[
    Height Field Fluid Simulation in Roblox Studio
    Based on "Height Field Fluids" by Matthias Müller

    This script simulates the shallow water equations over a terrain heightmap.
    The simulation includes terrain interaction, fluid dynamics, numerical stability enhancements, and visualization.
]]

-- User-adjustable parameters
local GRID_WIDTH = 100      -- Number of grid cells in x-direction
local GRID_HEIGHT = 100     -- Number of grid cells in y-direction
local DX = 4                -- Cell size in x (studs)
local DY = 4                -- Cell size in y (studs)
local OFFSET_X = 0          -- World offset in x-direction (adjust to match terrain position)
local OFFSET_Z = 0          -- World offset in z-direction (adjust to match terrain position)
local GRAVITY = 100        -- Acceleration due to gravity (studs/s^2)
local CFL_NUMBER = 0.5      -- Courant number for stability (<= 1)
local WATER_ADDITION_RATE = 500  -- Rate at which water is added at the source (studs per second)
local WATER_SOURCE_RADIUS = 2   -- Radius of water source area (in cells)

-- Stability parameters
local MAX_WATER_HEIGHT = 100    -- Maximum allowable water height
local MAX_SLOPE = 1             -- Maximum allowable slope between cells
local DAMPING_COEFFICIENT = 0.1 -- Damping coefficient for fluxes
local DIFFUSION_COEFFICIENT = 0.05  -- Diffusion coefficient for water heights
local EPSILON = 1e-6            -- Small epsilon to prevent division by zero

-- Data structures
local H_Terrain = {}    -- Terrain height
local H_Water = {}      -- Water depth
local E = {}            -- Water surface elevation (h + z)
local Qx = {}           -- Flux in x-direction
local Qy = {}           -- Flux in y-direction
local WaterParts = {}   -- Visual representation of water

-- Initialize data structures
for i = 1, GRID_WIDTH + 2 do  -- Added buffer cells for safety
	H_Terrain[i] = {}
	H_Water[i] = {}
	E[i] = {}
	Qx[i] = {}
	Qy[i] = {}
	for j = 1, GRID_HEIGHT + 2 do
		H_Terrain[i][j] = 0
		H_Water[i][j] = 0
		E[i][j] = 0
		Qx[i][j] = 0
		Qy[i][j] = 0
	end
end

-- Function to sample terrain height using raycasts
local function SampleTerrainHeight(i, j)
	-- Convert grid indices to world coordinates
	local worldX = (i - 1) * DX + OFFSET_X
	local worldZ = (j - 1) * DY + OFFSET_Z
	local origin = Vector3.new(worldX, 1000, worldZ)   -- Start raycast from above
	local direction = Vector3.new(0, -2000, 0)         -- Raycast downwards

	-- Raycast parameters
	local params = RaycastParams.new()
	params.FilterDescendantsInstances = {workspace.Terrain}  -- Whitelist terrain
	params.FilterType = Enum.RaycastFilterType.Whitelist
	params.IgnoreWater = true  -- Ignore water parts

	local rayResult = workspace:Raycast(origin, direction, params)
	if rayResult then
		local terrainHeight = rayResult.Position.Y
		return terrainHeight
	else
		-- Default terrain height if nothing is hit
		return 0
	end
end

-- Generate the terrain heightmap
function GenerateTerrainHeightmap()
	for i = 1, GRID_WIDTH + 2 do
		for j = 1, GRID_HEIGHT + 2 do
			H_Terrain[i][j] = SampleTerrainHeight(i, j)
			H_Water[i][j] = 0
			E[i][j] = H_Terrain[i][j] + H_Water[i][j]
		end
	end
end

-- Water Source
local WaterSource = {
	x = math.floor(GRID_WIDTH / 2) + 1,
	y = math.floor(GRID_HEIGHT / 2) + 1,
	rate = WATER_ADDITION_RATE  -- Water addition rate
}

function AddWater(deltaTime)
	local radius = WATER_SOURCE_RADIUS
	local totalCells = (2 * radius + 1) ^ 2
	for i = WaterSource.x - radius, WaterSource.x + radius do
		for j = WaterSource.y - radius, WaterSource.y + radius do
			if i >= 1 and i <= GRID_WIDTH + 2 and j >= 1 and j <= GRID_HEIGHT + 2 then
				-- Add water to the cell
				H_Water[i][j] = H_Water[i][j] + (WaterSource.rate * deltaTime) / totalCells
				-- Update water surface elevation
				E[i][j] = H_Terrain[i][j] + H_Water[i][j]
			end
		end
	end
end

-- Compute fluxes with slope limiting and water height limits
function ComputeFluxes(deltaTime)
	-- Compute fluxes in x-direction
	for i = 1, GRID_WIDTH + 1 do
		for j = 1, GRID_HEIGHT + 2 do
			local hL = H_Water[i][j]
			local hR = H_Water[i + 1][j]
			local hAvg = math.min((hL + hR) / 2, MAX_WATER_HEIGHT)
			local deltaE = E[i + 1][j] - E[i][j]
			-- Limit the slope
			local maxDeltaE = MAX_SLOPE * DX
			deltaE = math.clamp(deltaE, -maxDeltaE, maxDeltaE)
			Qx[i][j] = -GRAVITY * deltaTime * hAvg * (deltaE / DX)
			-- Apply damping to fluxes
			Qx[i][j] = Qx[i][j] * (1 - DAMPING_COEFFICIENT)
		end
	end

	-- Compute fluxes in y-direction
	for i = 1, GRID_WIDTH + 2 do
		for j = 1, GRID_HEIGHT + 1 do
			local hB = H_Water[i][j]
			local hT = H_Water[i][j + 1]
			local hAvg = math.min((hB + hT) / 2, MAX_WATER_HEIGHT)
			local deltaE = E[i][j + 1] - E[i][j]
			-- Limit the slope
			local maxDeltaE = MAX_SLOPE * DY
			deltaE = math.clamp(deltaE, -maxDeltaE, maxDeltaE)
			Qy[i][j] = -GRAVITY * deltaTime * hAvg * (deltaE / DY)
			-- Apply damping to fluxes
			Qy[i][j] = Qy[i][j] * (1 - DAMPING_COEFFICIENT)
		end
	end
end

-- Update water heights and apply numerical diffusion
function UpdateHeights()
	local newH_Water = {}
	for i = 1, GRID_WIDTH + 2 do
		newH_Water[i] = {}
		for j = 1, GRID_HEIGHT + 2 do
			newH_Water[i][j] = H_Water[i][j]
		end
	end

	-- Update water heights based on fluxes
	for i = 2, GRID_WIDTH + 1 do
		for j = 2, GRID_HEIGHT + 1 do
			local fluxInX = Qx[i - 1][j]
			local fluxOutX = Qx[i][j]
			local fluxInY = Qy[i][j - 1]
			local fluxOutY = Qy[i][j]

			local netFlux = (fluxInX - fluxOutX) / DX + (fluxInY - fluxOutY) / DY
			newH_Water[i][j] = H_Water[i][j] + netFlux

			-- Prevent negative water height
			if newH_Water[i][j] < 0 then
				newH_Water[i][j] = 0
			elseif newH_Water[i][j] > MAX_WATER_HEIGHT then
				-- Limit maximum water height
				newH_Water[i][j] = MAX_WATER_HEIGHT
			end
		end
	end

	-- Apply numerical diffusion
	for i = 2, GRID_WIDTH + 1 do
		for j = 2, GRID_HEIGHT + 1 do
			local laplacian = H_Water[i -1][j] + H_Water[i +1][j] + H_Water[i][j -1] + H_Water[i][j +1] - 4 * H_Water[i][j]
			newH_Water[i][j] = newH_Water[i][j] + DIFFUSION_COEFFICIENT * laplacian
			-- Ensure non-negative water height
			if newH_Water[i][j] < 0 then
				newH_Water[i][j] = 0
			end
		end
	end

	-- Update water heights and surface elevations
	for i = 1, GRID_WIDTH + 2 do
		for j = 1, GRID_HEIGHT + 2 do
			H_Water[i][j] = newH_Water[i][j]
			E[i][j] = H_Terrain[i][j] + H_Water[i][j]
		end
	end
end

-- Enforce boundary conditions
function EnforceBoundaryConditions()
	-- Zero-flux boundary conditions
	for j = 1, GRID_HEIGHT + 2 do
		Qx[1][j] = 0
		Qx[GRID_WIDTH + 1][j] = 0
	end
	for i = 1, GRID_WIDTH + 2 do
		Qy[i][1] = 0
		Qy[i][GRID_HEIGHT + 1] = 0
	end
	-- Reflective boundary conditions
	for i = 1, GRID_WIDTH + 2 do
		H_Water[i][1] = H_Water[i][2]
		H_Water[i][GRID_HEIGHT + 2] = H_Water[i][GRID_HEIGHT + 1]
		E[i][1] = H_Terrain[i][1] + H_Water[i][1]
		E[i][GRID_HEIGHT + 2] = H_Terrain[i][GRID_HEIGHT + 2] + H_Water[i][GRID_HEIGHT + 2]
	end
	for j = 1, GRID_HEIGHT + 2 do
		H_Water[1][j] = H_Water[2][j]
		H_Water[GRID_WIDTH + 2][j] = H_Water[GRID_WIDTH + 1][j]
		E[1][j] = H_Terrain[1][j] + H_Water[1][j]
		E[GRID_WIDTH + 2][j] = H_Terrain[GRID_WIDTH + 2][j] + H_Water[GRID_WIDTH + 2][j]
	end
end

-- Rendering
function InitializeWaterParts()
	local WaterModel = Instance.new("Model")
	WaterModel.Name = "Water"
	WaterModel.Parent = workspace

	for i = 1, GRID_WIDTH + 2 do
		WaterParts[i] = {}
		for j = 1, GRID_HEIGHT + 2 do
			local part = Instance.new("Part")
			part.Size = Vector3.new(DX, 1, DY)
			part.Anchored = true
			part.CanCollide = false
			part.Material = Enum.Material.Water
			part.Color = Color3.fromRGB(0, 162, 255)
			part.Transparency = 0.5
			part.Parent = WaterModel
			WaterParts[i][j] = part
		end
	end
end

function RenderSimulation()
	for i = 2, GRID_WIDTH + 1 do
		for j = 2, GRID_HEIGHT + 1 do
			local waterDepth = H_Water[i][j]
			local terrainHeight = H_Terrain[i][j]
			local part = WaterParts[i][j]

			if waterDepth > 0.01 then
				-- Position the water part
				part.Position = Vector3.new(
					(i - 1) * DX + OFFSET_X,
					terrainHeight + waterDepth / 2,
					(j - 1) * DY + OFFSET_Z
				)
				-- Adjust the size (height)
				part.Size = Vector3.new(DX, waterDepth, DY)
				part.Transparency = 0.5
			else
				-- Hide the part if no water
				part.Size = Vector3.new(0, 0, 0)
				part.Transparency = 1
			end
		end
	end
end

-- Initialize the simulation
GenerateTerrainHeightmap()
InitializeWaterParts()

-- Main simulation loop
while true do
	-- Compute maximum water depth for time step calculation
	local maxWaterDepth = 0
	for i = 2, GRID_WIDTH + 1 do
		for j = 2, GRID_HEIGHT + 1 do
			if H_Water[i][j] > maxWaterDepth then
				maxWaterDepth = H_Water[i][j]
			end
		end
	end
	if maxWaterDepth < EPSILON then
		maxWaterDepth = EPSILON
	end

	-- Compute time step based on CFL condition
	local deltaTime = CFL_NUMBER * math.min(DX, DY) / (math.sqrt(GRAVITY * maxWaterDepth) + EPSILON)
	deltaTime = math.min(deltaTime, 0.01)  -- Cap deltaTime to a safe maximum

	-- Simulation steps
	EnforceBoundaryConditions()
	ComputeFluxes(deltaTime)
	UpdateHeights()
	AddWater(deltaTime)
	RenderSimulation()

	-- Wait for the next time step
	wait(deltaTime)
end



And add these things: The code you've provided covers several important aspects of a height field fluid simulation, but based on the "Height Field Fluids" paper by Matthias Müller, there are a few key features or improvements that could be added or refined to better match the methodology described in the paper:

### 1. **Flux Limiting for Numerical Stability**
   - In the paper, flux limiting is a crucial technique to ensure numerical stability, especially when water flow between cells can result in steep slopes. While you've implemented a basic version with clamping, the paper might propose a more sophisticated approach, such as slope limiting using Total Variation Diminishing (TVD) schemes to avoid non-physical oscillations near sharp changes in water height.

### 2. **Advection**
   - The simulation currently lacks an explicit advection step to move water along the flow field. The advection process ensures that the water volume is transported from one cell to another in a manner consistent with fluid velocity. Proper advection can improve the realism of water movement and should be based on the flow field (velocities in X and Y directions).

### 3. **Detailed Boundary Handling**
   - The current boundary conditions are relatively simple, using reflective conditions (zero-flux boundaries). In the paper, there might be more advanced boundary conditions such as absorbing boundaries to prevent reflections or inflow/outflow boundaries to simulate water sources and sinks more realistically.

### 4. **Sub-Grid Terrain Slope Calculation**
   - In Müller’s approach, terrain slopes can be computed using finer sub-grid calculations to capture the precise interaction between the water and terrain. This allows for more accurate fluid flow over sloped terrain. The slope calculation can be extended to use higher-order methods to avoid discontinuities in steep terrain areas.

### 5. **Wave Propagation with Smoothing**
   - In order to better simulate wave propagation, smoothing filters can be applied to water heights. The paper may describe how to dampen high-frequency noise in the simulation while preserving important wave features, especially when simulating larger water bodies.

### 6. **Handling of Wet/Dry Boundaries**
   - Wetting and drying algorithms ensure that water doesn’t unrealistically pool in cells that become dry. A specialized approach for this is often necessary to manage regions where water height becomes near zero. The handling of these areas is critical to prevent numerical instability.

### 7. **Advanced Diffusion Handling**
   - The diffusion in your code is implemented with a simple laplacian formula, but the paper may introduce more complex diffusion schemes that ensure more realistic water distribution, especially for turbulent flows or when water interacts with very steep terrain. Numerical dissipation might need better handling.

### 8. **Efficiency Considerations (Parallelism)**
   - Müller’s paper may include sections on efficiency and parallelism that are essential for large-scale fluid simulations. While your current script works, optimization through parallelism or multi-threading (for example, dividing the grid into chunks for more efficient updates) can drastically improve performance.

### 9. **Visualization and Detail Improvements**
   - You could add features like foam, splash particles, or different visual cues for turbulent water to enhance realism. The paper may discuss using additional visual effects to enhance the perception of water motion, like adjusting color and texture based on water velocity or depth.

### 10. **Viscosity**
   - Depending on the simulation goals, viscosity could be an important factor in controlling how "thick" or "thin" the fluid behaves. Implementing a variable viscosity model, as suggested in the paper, could improve realism for various types of fluids (e.g., mud versus clear water).

### 11. **Water Conservation**
   - Ensure that the total amount of water in the system is conserved unless explicitly removed (for example, by a sink boundary). The paper may provide a more robust method to account for water conservation and ensure no volume loss or gain due to numerical inaccuracies.

### 12. **Advanced Source Terms**
   - The water addition through sources could be improved to better match realistic inflow dynamics. In Müller’s paper, there might be more sophisticated approaches to handling dynamic water sources such as rain or rivers entering the simulation from the boundary.

### 13. **Pressure Solvers**
   - For larger bodies of water or more complex scenarios, Müller’s paper could suggest the need for pressure solvers to handle incompressibility, especially in 3D extensions of the height field method. Although this script is 2D, pressure can still play a role in ensuring more realistic behavior of dense fluid.

### Conclusion:
The current script implements a basic height field fluid simulation, but some of the more advanced features discussed in the paper could improve stability, realism, and performance, such as:
- Flux limiting
- Advanced advection schemes
- Detailed boundary handling
- Handling wet/dry boundaries
- Improved terrain interaction.

Enhancing these areas will bring your simulation closer to the methodology proposed in the "Height Field Fluids" paper.